Monetary Policy, the Financial Accelerator model and Asset

Price bubbles ∗

Eddie Gerba†

University of Kent

This version: April 2011

Preliminary and Incomplete: Comments welcome

Abstract

US net worth and leverage in both the corporate and the household sectors became much more

pro-cyclical and volatile in the 2000s, while financial spreads became more counter-cyclical. We

assess the role of monetary policy in contributing to this change. We use an extension of the BGG

model and analyze the role that neglecting asset prices might have had in the most recent business

cycle. Our results suggest when policy responds to an identified bubble it naturally increases

welfare relative to alternatives. Additionally, we measure the extent to which policy can feed a

bubble by further loosening the credit market frictions that entrepreneurs face. We also gouge the

relative losses from acting against asset prices when there is no bubble and formulate an asset price

targeting frontier.

Keywords : Financial frictions, asset price bubbles, US economy, monetary policy, model uncertainty

JEL codes : E3, E4, E5, C1, C6

∗I would like to thank Profossor Jagjit Chadha for his great support and supervision I also would like to warmly thankEvren Caglar for assisting me in simulating the model, and James Warren, Alex Waters and Jack Meaning for their usefulcomments.†School of Economics, Keynes College, University of Kent, Canterbury, Kent, CT2 7NP, UK. E-mail: eg229@kent.ac.uk

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1 Introduction

The recent financial turmoil has revived the debate on whether central banks should respond to asset

price movements. Previous studies in this literature had shown that strong inflation targeting was

sufficient in bringing down the stock market booms and control the economy, often relying on the

premises that the expansionary effects of asset price bubbles are well captured by inflation and output.

Moreover, since bubbles are hard to observe, a policy maker that tries to target them might increase

the range of instability of models (Bullard and Shaling (2002)). But recent empirical studies from

the current financial crisis have pointed out that the unusually low Federal Funds rate during the

entire boom period in the 2000’s contributed largely to feeding the stock-and property market bubbles.

Capitalizing on these new observations, our paper returns to the pre-crisis debate, and in a model

with financial imperfections, investigates what role neglecting asset prices from monetary policy play

in a bubble, and how much less of a bubble might there have been if alternative policy rules had been

employed.

In models where financial frictions are absent, a policy responding strongly to inflation is sufficient, since

suppressing inflation stabilizes the markup. The only distortion in this type of model is the variation

in markup (Gilchrist and Saito (2006)). However, financial frictions are a major cause of economic

fluctuations. We therefore concentrate our analysis on the financial accelerator model of Bernanke-

Gertler-Glichrist, BGG (1999), where credit market imperfection are a key driving force of the business

cycle fluctuations. 1 We extend the canonical model to incorporate an exogenous asset price bubble,

and make corporate investment contingent on market asset prices, basing our latter modification on

studies made by (Bond and Cummins (2001), and Dupor (2005) who show that market asset prices

directly influence managerial valuations of capital, and hence their investment decisions. We calibrate

and simulate the model, and study the way in which pro-cyclical monetary policy influences asset price

booms. We study the responses of the economy under an accommodative-and an aggressive monetary

policy conduct. Once the economic responses have been established, we go on to perform a detailed

welfare analysis. In particular, we are interested in finding the optimal monetary policy in welfare terms

under alternative calibrations of the bubble process.

However, we recognize that it might be difficult to identify a bubble. It is very difficult for a policy

maker to know whether a given change in asset values results from fundamental factors, non-fundamental

factors, or both. The risks embedded in contracting healthy economic growth driven by fundamentals

by ’overreacting’ in the response of interest rates to booms, as well as the historically relevant risk

of panic after the bubble has exploded are high, and hence the problems in targeting the asset prices

are not irrelevant. Studies such as the one put forward by Ben Bernanke and Mark Gertler (2000)

illuminate this problematic and argue for a strong inflation targeting. We incorporate this problem

into our welfare analysis, and compare the welfare losses when the central bank is able to identify a

bubble, and when it is not. More specifically, we wish to study the losses generated when the monetary

authority does not target an asset price bubble because it id not identify one, and there is a stock

market boom (Type I error), and when the authority thinks there is a bubble and targets it when in

reality there is no bubble (Type II error), and compare those to the respective losses generated when

the authority targets a bubble and there is one in the economy, and vice versa. We do this for multiple

scenarios of the bubble process. In this way we can identify the forecast errors in each type of monetary

policy rule, and decide upon which targeting is optimal including the forecast errors.

1While more elaborated financial DSGE models have been published during the past few years, they all have thefinancial accelerator mechanism of BGG as their fundamental friction, and for technical simplicity, we opt for this modelin our analysis.

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Other studies in this literature have pointed in the opposite direction. Cecchetti et al (2000, 2003), and

Mussa (2002) argue that an asset price targeting is not only beneficial, but also necessary if a policy

maker has a stable economic development as its goal. The argument, largely taken from control theory,

has been that if asset prices are macroeconomic state variables, they should be responded to as any

other state variable. The measurement error inherent in the identification of those bubbles, and in the

origins of the bubble shocks may attenuate the responses to asset price movements, requiring more focus

in filtering correctly the information. However, it does not mean that it is optimal to fail to respond to

such movements.

More recent studies take a middle ground position in this debate, and argue that the monetary authority

should, in addition to inflation, react to asset price differences only when the bubble persistence is small,

or when the bubble shock is large (Tetlow (2005)). Alternatively, they should react to the gap between

the observed and the potential level of asset prices (Gilchrist and Saito (2006)) since bubbles are usually

associated with technology growth. In this setting, asset prices reflect expectations about the underlying

changes in the trend growth rate of technology, meaning that changes in asset prices are a result of

changes in fundamental, as well as bubble factors.

The paper makes three contributions. Using the methodology of Stock and Watson (1998), we extend

the dataset to include more financial variables, as well as a longer time series (1953:Q1 to 2010:Q3),

and examine the macro-financial linkages in the US since the post-war period. We show that the U.S.

economy changed structurally around the millennium shift, and became more volatile since 2000. We

attribute this shift to the stock-and property market booms. This was reflected in an increased volatility

and procyclicality of both firm and household net worth and leverage, as well as in asset prices. During

the same period, major financial spreads became more countercyclical, and were as volatile during the

past six years as during the turbulent years of 1970’s. Second, we show how procyclical monetary policy

contributes to an asset price boom. Third, our welfare analysis results illustrate how much better the

economy would be if an optimal policy rule had been applied, i.e. if the market asset prices had been

targeted.

The rest of the paper is outlined as follows. Following the introduction, we depict the model we use

in this paper, and extend it to accommodate for an asset price bubble. A more thorough theoretical

foundation is outlined in Annex A. Section III introduces the financial stylized facts over U.S. business

cycles for the last six decades. it is followed by an analysis of model impulse responses to a series of

shocks, as well as a matching moment analysis of the model and data second moments. In section V

we outline and perform our welfare experiments. The sixth and final section concludes.

2 Outline and discussion of the model

One model that has largely been used by Central Bankers to analyze the effects of asset price booms

on credit markets and on real activity has been the Kansas City version of the Bernanke, Gertler and

Gilchrist (1999) model (hence after, KC-BGG). From first principles, the model enables us to study how

large increases (decreases) in equity prices work to ease (tighten) credit constraints faced by investors,

and thus expand (shrink) the economy. The model is an extension of the canonical BGG (1998) model

to allow for an asset price bubble which is independent from the fundamental (economic) value. As a

result, both the financial and economic variables depend on the performance of the stock market, which

in turn depends on the financial constraints that investors face. Apart from this financial accelerator

mechanism, the rest of the framework is structured as a conventional New-Keynesian DSGE model.

The model is moreover discrete. It means that the economy starts off with some initial level of capital

stock, and net wealth of entrepreneurs, and in all the subsequent periods entrepreneurs purchase their

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entire capital stock at the beginning of each period. In addition, only one-period financial contracts are

considered in this model that are terminated at the end of each period. A more detailed description of

the model can be found in Annex A, with the corresponding optimization problems of each agent, as

well as the log-linearized version of the general equilibrium model.

There are five types of agents in this model: households, entrepreneurs, retailers, intermediaries and a

government sector. Except for entrepreneurs, the decision problems of the other agents are conventional.

Households, who live forever, decide in each period how much to work, and either consume, or save

their earnings in bank deposits, which turns them into lenders in this model. Their utility is assumed

to be separable over time and over leisure, consumption, and real money balances.

Finitely-living entrepreneurs, on the other hand, decide in each period to either consume, or invest their

profits in stock markets. They hold only limited internal financing, so in order to finance the rest of

their investment demand, they are obliged to take loans from banks. Despite them wanting to borrow

the maximum amount, because of information asymmetries, and incentive-and monitoring problems

between lenders and borrowers, they are only allowed to borrow as much as their net wealth is worth.

Therefore, movements on stock markets do not only determine entrepreneurs’ net wealth, but also the

constraints they face on the credit market. Because external financing is costly, a spread between raising

the financing externally and the internal wealth arises, called the external finance premium (hence after

EFP). This premium for external finances affects positively the overall cost of capital, and thus inversely

the real investment decisions of firms. Hence, a stock market bubble will lead to pronounced procyclical

investment and spending behavior, whilst busts will lead to pronounced recessions in both the credit

market and the general economy. As a result, the financial accelerator enhances the effects of initial

shocks to the economy.

There are two types of capital prices in this model. The fundamental value of assets is defined as the

present value of the dividends that the capital is expected to generate:

Qt = Et

∞∑i=0

[(1− δ)iDt+i+1/

i∏j=0

Rqt+j+1

](1)

where Qt is the fundamental asset price at date ’t’, Et indicates the expectation as of period ’t’, δ is

the depreciation rate of capital, Dt+1+i are the dividends paid out, and Rqt+1 is the stochastic gross

discount rate at ’t’ for dividends received in period ’t+1’. Market asset price, the other capital price, is

tightly related to asset price bubbles. A bubble is considered to exist whenever the market share price,

∆t is above the fundamental asset price,Qt, i.e. when ∆t −Qt 6= 0. If a bubble exists, it persists with

probability, p, and conditional on its persistence, it grows at a rate a/p, where a is the expected growth

rate of the bubble:

∆t+1 −Qt+1 =a

p(∆t −Qt)R

qt (2)

,where Rqt is the stochastic gross discount rate in ’t’ of dividends received in ’t+1’. If the bubble does

not exist, then the above expression collapses to zero, and ∆t+1 = Qt+1. The bubble is parametrized

such that it lasts for five periods in the benchmark calibration, and since a is restricted to be less than

unity, the discounted value of bubble converges to zero with a fixed probability. Nevertheless, later on

in our welfare analysis experiments, we will consider three cases of bubbles by calibrating the parameter

a in different ways. the three type of bubble processes we will analyze is a mean-reverting bubble that

only lasts for five periods, and dies out. For this processes, we calibrate the value of a to 0.98. The

second type of bubble is a rational bubble in the sense of Blanchard and Watson (1982), that also lasts

for five periods. For this, we set the parameter a equal to 1. Finally, to contrast the previous cases to

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an economy which lacks a bubble, we calibrate the parameter a to 1.025, and hence set ∆t −Qt = 0.

The market value of shares, which is inclusive of the bubble is defined according to the following process:

∆t = Et

∞∑i=0

[(1− δ)iDt+i+1/

i∏j=0

R∆t+j+1

](3)

where R∆t+1 is the future return on stocks, and is related to the return on fundamentals by:

R∆t+1 = Rq

t+1[b+ (1− b)Qt

∆t

] (4)

where b is a parameter on the bubble governed by the total period of time the bubble persists, and the

depreciation rate of capital. The bubble is assumed to persist with a fixed probability into the next

period. However, if not bubble exists, then the return on shares is the same as return on fundamentals,

i.e.:

R∆t+1 = Rq

t+1 (5)

Besides the bubble process, a second assumption is introduced regarding the investment decision of

entrepreneurs. Following the empirical studies made by Bond and Cummins (2001), and Dupor (2005)

that managers make their investment decisions based on stock market developments, as well as Gilchrist

and Saito (2006) who integrate this important remark into their version of the BGG model, we assume

that entrepreneurs value their capital, and subsequently invest on capital markets based on the market

price of assets (i.e. including the exogenous bubble component of asset prices), conditional on the cost

of external credit. Since the return on investment will depend on the rate of return of shares, the

balance sheet condition of entrepreneurs will depend on the share price, rather than the fundamental

price as in the original model. This extension can visually be represented by Figure 1. The diagram

represents the market for external finances that entrepreneurs face, and combines both the logic from

the canonical BGG model, as well as the KC extension. For financing needs up to ′IF ′, the debtor

can use internal funds to finance his investments, at an opportunity cost equal to the risk-free rate R.

Without informational frictions, the debtor would demand EF −IF of external financing, at an interest

rate of R. But, since the model incorporates agency costs, for financing above IF , the creditor is not

prepared to supply loans at this risk-free rate, and instead charges a premium. Because of increasing

default rates with higher external financing, the premium also increases. Hence, creditors require ,ore

compensation, and S is upward sloping beyond IF . The equilibrium is at EF − IF . As interest rates

increase, the premium also increases because of the lower present discounted value of collateral, or

reduced cash flows. The new market rate charged is thus R′, and the supply schedule becomes more

inelastic, represented by S ′. With the assumption of share price-based investment in the KC extension

of the model, however, the constraint in supply of funds that entrepreneurs face is relaxed again, as the

value of collateral is increased (or the cash flows), and hence a lower market rate is charged for loans,

R′′. At the same time, creditors are more relaxed about the financial conditions of debtors, and since

they supply more loans on the market, the supply schedule becomes less steep than S ′. The gradient

of the supply returns to its initial value, even with a higher interest rate, as the market conditions in

external finances have improved.

Hence for the lending constraint of the extended model, an unexpected rise in price of shares above its

fundamental driven by the bubble process will not only incentivize a rise in investment from the internal

funds, but also relax the collateral constraint that they face on their loans since the probability of default

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is reduced, allowing them to borrow more, and hence boost the investment even further. Since share

prices are assumed to move independently from the fundamental price, and they are exogenously defined,

the volatility of the aggregate economy will be enhanced, and the initial shocks will be propagated and

amplified with a stronger force compared to the canonical version.

As a last remark on asset price booms, there is also a wealth effect on consumption from a stock market

boom. However, following Ludvigson and Steindel (1999), the model is parametrized so that these

effects are of second-order importance, just as in KC.

Banks, retailers and government have a secondary role in this model. Banks, acting as intermediaries

between lenders and borrowers on the credit market, can be viewed as a veil. They assist in matching

lenders and borrowers, and monitoring borrowers, but they lack an explicit decision problem, so the

actual constituents of the financial sector in this model are lenders, who are risk averse and require a

return on their lending equal to the risk-less rate Rt+1, and borrowers, who are risk neutral and face

a stochastic return on their equity equal to R∆t+1. If the return on capital markets in the next period

turns out to be lower than expected, the loan rate on the credit market will go up due to an increased

probability of default on loans that entrepreneurs have taken. That is why the loan rate will adjust

countercyclically in this model.

Retailers are simply included in this model to introduce the nominal stickiness standard in the DKN-

framework. They buy intermediate goods produced by entrepreneurs, differentiate them costless and

sell them to households in a monopolistically competitive market. They set their prices following the

common Calvo process, and retailers that set their price in period ’t’ are assumed to do so before any

aggregate uncertainty has been realized. Because of this heterogeneous price setting ability, the model

assumes that there is a continuum of retailers in the interval [0, 1]. The profits they earn are transferred

in lump-sum to households.

Finally the government sector, comprised of both a Central Bank (CB) and a Government, acts as

an economic stabilizer. The CB, which actively adapts the policy rate according to general economic

outlook, uses a fairly standard Taylor rule in setting their target rate. Note here, however, that monetary

policy has an additional transmission channel trough which it can affect the real activity. Apart from

the standard spending channel, it moreover affects the balance sheet of borrowers such that a rise in

interest rates, reduces asset prices, deteriorating the financial conditions of borrowers, and thus reducing

firm investment who then face a higher EFP. Hence, CB’s ability to manage the economy is stronger in

this case compared to a standard DNK model. The government, on the other hand, plays a marginal

role in this model (except for the case of a fiscal policy shock), and runs a balanced budget in each

period.

3 Financial facts on the US Business Cycle

3.1 Stylized facts

Before we go on to analyze the impact of asset price booms in our model, let us first have a look at

what US data tells us aboutthe financial and economic developments since post-World War II period.

The majority of the firm and household financial data is taken from Reuter’s EcoWin database except

for the data on debt outstanding and consumption which are taken from Federal Reserve Bank of St.

Louis database. General macroeconomic data is also taken from the latter source (exception is the

TWEI series which is taken from Retuer’s EcoWin database), whilst data on financial rates and spreads

comes from multiple sources. 3-month Commercial Paper, and effective fund rates data come from the

IMF database, while Moody’s 30-year seasonal AAA, BAA and Government Benchmark rates come

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from Credit Market Trend Service. The rest of the rates, 3-month LIBOR rate, 3-month Treasury-bill

10-year Treasury-bond, 3-month Prime, 3-month Deposit (Euro-Dollar deposits), and 3-month Com-

mercial paper (AA-Non financial discounted) rates are extracted from Reuter’s EcoWin.

Expanding on Stock and Watson’s (1998) US long-run data analysis, we look at the quarterly firm

and household financial data as well as the general economic one from 1953-20102 to explore the nexus

between corporate and household financial performance and the real activity. We expand Stock and

Watson’s original study by both extending the time series data to include the 2000’s as well as the

number of financial variables analyzed. We include more financial spreads and more variables on cor-

porate and household finances. However, our main focus is on the variables which are essential for

the KC model. In addition, the data statistics are divided into two periods, 1953-2000 (pre-2000), and

2000-2010 (post-2000) because we find a structural break in majority of the data around the millennium

shift. Our distance matrix based tests of Box-Bartlett and Wald all confirm this observation. 3

Table 3 presents HP-filtered4 US corporate-,and household credit data for the period 1953:Q1-2010:Q4.

Both the statistics on the relative standard deviation with respect to output’s, and the correlation with

output of each variable are included. Graphs 2 to 4 depict the evolution of these variables included in

Table 3. All corporate data excludes the financial and farming industries, since those are structurally

different from the remaining corporate sector. Firm net worth at historical cost is defined as total firm

equity minus debt with assets recorded at their historical purchase value, while net worth at market

value is the same variable but with assets recorded at their current market value on their balance sheets.

Corporate profits are recorded net of taxes, and are inventory value-and capital consumption adjusted.

Debt outstanding is defined as unpaid debt in each quarter, including the interest on that debt, and is

denominated in nominal terms.

Most of the private sector variables included are procyclical. The only exception is corporate net worth

at market value, which lags considerably the GDP series during the 1970’s, hence why we get a negative

correlation overall during that period. But, if we calculate the same correlation statistic for net worth

(at both historical cost and market value) excluding the period 1969-83,5 the coefficient turns positive

for net worth at market value, and more positive for net worth at historical cost. The majority of

the variables are also more volatile than GDP, with a relative standard deviation of 25% to 7 times

higher than the GDP’s. The only exception is personal disposable income that is approximately 25%

less volatile than output.

Table 5 shows HP-detrended6 US key macroeconomic variables; output, investment, consumption, trade-

weighted exchange index (TWEI), government spending, and inflation. Graphs 3 to 5 depict the evo-

lution of these variables included in Table 5. Consumption is defined as private consumption of non

durables, TWEI is in real terms and includes most US-trading partners’ currencies, and government

spending includes real government consumption expenditure, and gross public investment. In lines with

the common wisdom, investment and consumption are procyclical, whilst government spending and

the TWEI are countercyclical. Investment and the TWEI are more volatile than output, consumption

and government spending are approximately as volatile as output, whilst inflation is the least volatile.

Investment stays highly correlated with GDP throughout the whole sample period (reaching almost a

2Except for the data on 10-year T-bond rate, and 3-month T-bill rate that are only available from fourth quarter of1954, Prime rate-spread from fourth quarter of 1955, Fund rate that is only available from second quarter of 1957, Depositspread being available from second quarter of 1971 Paper-Bill spread from a quarter later the same year, LIBOR-andTED-spreads from second quarter of 1975 and data on TWEI from second quarter of 1973.

3see table 3 for the null hypothesis and the respective test values4With λ = 16005Which is dominated by the turbulent period marked by the collapse of Bretton-Woods, the two oil-shocks, and

stagflation in US6With λ = 1600

7

perfect correlation with output), but consumption became 6 times more correlated with output in the

post-2000 period, compared to the earlier.

A third group of data is the HP-filtered 7 quarterly financial rates and spreads included in Table 4 and

represented by Graphs 5 to 7. The long term credit spreads here are AAA-spread (30-year AAA-rate

minus 30-year Gov. Benchmark), BAA spread (30-year BAA-rate minus 30-year Gov. Benchmark) and

Corporate risk spread (30-year BAA-rate minus 30-year AAA-rate). The short term lending spreads are

LIBOR spread (3-month LIBOR-rate minus 3-month Fund rate), TED spread (3-month LIBOR-rate

minus 3-month T-bill rate), Prime spreads (3-month Prime-rate minus 3-month Fund rate, and 3-month

T-bill rate) and we include Paper Bill spread (3-month Commercial paper-rate minus 3-month T-bill

rate) as a measure of financial market liquidity. As expected, the three (risk-less) financial rates follow

generally well the output series. Turning to the spread variables, the long-term spreads are countercycli-

cal in the whole sample period. In particular, the high countercyclicality of the corporate risk spread

reflects the increased risk of default on lower grade bonds that builds up during recessions, that results

in a widening of spread between bonds with different quality ratings. At the same time, the spreads are

3 to 5 times smoother than output. On the other hand, the short-term spreads are more volatile, with a

relative standard deviation of between 34% and 65% of output’s standard deviation. Their correlations

with output are also weaker compared to the long-term spreads.

3.2 The structural change in the data

To investigate whether the US economy underwent a profound change around the millennium shift,

we performed the Box-Bartlett, and Wald structural break tests. The values are reported in table 3.

For both tests, we got a high statistic, which lead us to reject the null hypothesis that the data are

structurally equal in the 1953-2000 period compared to 2000-2010. A closer look at the graphs and

second moments of all our variables, moreover, confirms these results. The majority of the corporate

and household credit variables become much more procyclical and volatile in the post-2000 period

compared to the pre-2000 one. The only exceptions are net corporate profits and real disposable income

of households that became less procyclical in the second period. For the macroeconomic variables, US

investment, consumption and inflation became significantly more procyclical, while the trade-index

and public spending became considerably more countercyclical in the post-2000 period. At the same

time, US investment became 27% more volatile, and consumption twice as volatile in the latter period

compared to earlier. As a last set of variables, we see also a significant change in the rate and spread

data. The three risk-free rates became much more procyclical since 2000’s, while for the long-term

spreads, the negative correlation to output slightly falls for the post-2000 period and the volatility

increases around, and after the 2001 dot.com bust. In contrast, the volatility of the short-term spreads

included here decreases after 2000, but they remain as more volatile than the long-term spreads.

For prime rate spread, the significant decrease in volatility since the 1980’s suggests that the cost of

lending has been more than halved since the deregulation of the financial sector in the early 80’s. Paper

bill spread turned countercyclical in the post-2000 period, just as the deposit-and LIBOR spreads.

Moreover, the deposit-, LIBOR- and TED-spreads have their highest countercyclical peak since the

oil-shocks in 1979 in 2008, following the collapse of Lehman Brothers.

Summarizing the relevant U.S. stylized facts for the KC-model, it means that entrepreneurs’ net worth

in the model should be highly procyclical, and approximately 2 to 5 times more volatile than output.

Asset prices should exhibit an almost perfect correlation with output, and be as volatile as 6 to 11 times

the standard deviation of GDP. The same is true for investment series, but with a slightly lower volatility

7With λ = 1600

8

factor of 5 times the one for GDP. Leverage should also be highly procyclical, and approximately twice

as volatile as output, in particular for the post-2000 period. While less procyclical than the other

variables, consumption should in any case follow relatively well the output series, and be as volatile as

output, if not slightly more.

Finally, for the market rate of borrowing, or external finance premium as it is called in BGG, it should

be either weakly countercyclical, or countercyclical (depending on whether one models the lending as a

short-term, or a long-term contract), and significantly smoother than output.

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4 Quantitative analysis

4.1 Parameters and calibration

The model contains two types of parameters: Steady state parameters, that affect the steady state

relationships, and behavioral (remaining) parameters which change the behavior of the system once

their values are changed. Both type of parameters, together with the corresponding calibrations are

reported in Table 6.

The list in ’Variable’ specifies values of the parameters that we used to simulate the model. In what

follows, we mostly calibrate our model based on the parametrization given in the KC paper, and/or

canonical BGG model for parameters that were not reported in the former.

As a minor remark, for the parameter on net worth accumulation, the KC paper applies a slightly

different method from the original version. They multiply all of the determining variables of the net

worth evolution equation by Rq, the stochastic gross discount rate (which also appears as the denom-

inator in the evolution of bubble equation), while only a fraction of those determining variables are

multiplied by the steady state parameter in the original BGG model. Moreover, the authors do not

give an indication of the value of this parameter in any of the papers. Also, the Taylor rule is slightly

different in the KC version. Whilst in our simulated equations and in BGG (1998), the current nominal

rate is made contingent on the past interest rate and future inflation expectations, in the KC-BGG, it

is made contingent on the long-run interest rate. Therefore, the coefficient on the ’t-1’ interest rate is

missing in that case.

Comparing the different values of the parameters, the two parameters which considerably differ between

our simulation and the KC or BGG paper are σ and φ. For the intertemporal consumption elasticity

parameter, we chose the value 0.1 for this parameter in order to scale down the initial responses of

the model variables to an exogenous disturbance, without changing their path (i.e. we scale down the

magnitudes of the initial impulse responses). The same is true for our choice of φ. We have made

asset prices less responsive to the investment ratio, smoothing thus the responses of model variables

to an unexpected temporal asset price shock. To sum up, we have chosen those values to reduce the

sensitivity of the model responses since the authors of the model have pointed out that the drawback

with their calibration is the excessive responsiveness of investment and output to shocks.

As a last remark, let us comment on the parameter on net worth accumulation. In the original BGG

model, authors multiply the time ’t’ risk spread with a compounded parameter γRKN

(γ = 0.9728,

R = 1.01, and KN

= 2), giving a final value of 1.96. This parameter determines the impact of leverage

on the financial conditions of entrepreneurs in SS. The higher the parameter, the more exposed the

entrepreneurial sector is to capital market conditions, and the more accentuated are the business cycles.

4.2 Forcing variables

The four individual shocks that we are considering in our simulations are a 1 per cent unexpected

temporary shock to productivity, fiscal spending, monetary policy and lastly to an asset price bubble.

They are all described by a first-order autoregressive process. The asset price bubble shock we define

as a 1 per cent deviation of the market asset price from the fundamental price. To accommodate for

this shock, we introduce the shock matrix to the following system of linearized equations in the model:

E[rqt+1] = (1− ε)(xt + yt − kt−1) + εqt − qt−1 + Aut (6)

10

Et∆t+1 = φ(it+1 − kt+1)− Aut (7)

cet = ∆t +Kt+1 + Aut (8)

∆t − qt = (1− δbRq

)Et(∆t+1 − qt+1 + Aut) (9)

E[r∆t+1]− E[rqt+1] = −(1− b)∆t − qt + Aut (10)

E[r∆t+1] = (1− ε)(xt + yt − kt−1) + ε∆t −∆t−1 + Aut (11)

,where the matrix Aut represents the shock on asset price bubble at time ’t’ imposed on equations that

determine the asset prices. For some of the equations, however, the shock process is scaled down by a

parameter, since the variable which is shocked is also scaled by the same parameter.

The other three shocks in the model have a simpler process set-up. The monetary policy shock is an

exogenous disturbance to the nominal interest rate in the Taylor rule of the following form:

rnt = ρnrnt−1 + ρππt + ρyyt + εi,t (12)

where εi,t is the shock matrix at ’t’ to the nominal interest rate. In a similar manner, εgt is an unexpected

disturbance matrix to fiscal policy in period ’t’ in the resource constraint

yt =C

Yct +

Ce

Ycet +

I

Yit +

G

Ygt + εgt (13)

,and zt is an exogenous disturbance to productivity in period ’t’ in the production function

yt = zt + αkt−1 + (1− α)lt (14)

4.3 Simulations

We conducted a series of quantitative experiments in order to examine the dynamics of the key variables.

We simulated the four shocks described above, and compared the impulse responses for an economy

where the Central Bank applies an accommodative monetary policy rule (coefficient on the expected

inflation in the Taylor rule equal to 1.01) to those of an economy where the monetary authority applies

an aggressive Taylor rule ( coefficient on expected inflation equal to 2) in order to study the potential

of monetary policy in stabilizing the economy in this model.

By solving a linearized set of equilibrium conditions from the model, the impulse response functions

(IRFs) are computed. Figures ?? to 11 depict the IRFs to a 1 per cent unexpected temporary increase

in each exogenous variable.

We start off with an unexpected 1 per cent temporary rise in productivity (Figure 8). Consistent with

the conventional wisdom, a rise in productivity pushes down the inflation as the supply side is better

able to exploit their factors of production. While the fall in inflation in the accommodative case is

0.33 %, it is 0.20 % in the case of aggressive policy. In the latter case, the rise in productivity drives

the output up by 0.1 % and marginal costs down by 1.6 %, driving up the aggregate demand. As

11

a result, real rates fall by 0.15 %, and asset prices go up by 0.2 % above its steady state values, as

well as investment. Net worth increases by 0.25 %, reducing external finance premium by 0.05 %. On

the other hand, in the case of accomodative policy, the larger fall in inflation leads to a reduction in

aggregate demand. Whilst the fall in marginal costs by 2 % indicates that the initial response in output

is positive, because of a reduction in demand, reflected by just a marginal increase in asset prices and

investment equal to 0.0005 %, and an increase in household consumption by merely 0.002%, the total

effect on output is near to zero. Net worth in this case falls by 0.22 %, and the finance premium that

entrepreneurs face to rise by 0.015 %.

Turning now to shock in public spending in Figure 9, the economy marginally expands in the case of

an accommodative policy, and marginally contracts in the case of an aggressive policy. Whilst in both

cases, the fiscal expansion leads to an increase in output of 0.2 %, because the monetary authority

acts more aggressively to counteract the rise in inflation in the case of aggressive monetary policy, this

leads to a rise in real rate of 0.015%. Conversely, for the loose policy case, the economy marginally

expands. Looking at the accomodative case, the shock causes a rise in asset prices (both the market

and the fundamental) and investment by 0.003 %, pushing up net worth of entrepreneurs by 0.025 %,

and thus the external finance premium down by 0.001 %. This results in entrepreneurial level of debt

to rise by 0.002%. This leads to a rise in inflation of 0.035 %, and a rise in the policy rate by 0.033 %

to dampen the rising inflation. For the aggressive case, the responses in asset prices and investment are

to decline by 0.03 %, pushing down net wealth by 0.02 %, the external finance premium up by 0.005

% and the level of debt down by 0.06 %. Entrepreneurial consumption is also reduced by 0.03 % and

household consumption by 0.0035 %. Because of the more aggressive policy, the initial inflation only

rises by 0.02 %, and falls to its initial steady state level after 8 periods (whilst it takes it 20 periods for

the accomodative case).

For a positive shock to the policy rate, Figure 10, aggregate demand contracts in both policy cases.

However, the contraction is smaller for the aggressive policy. A 0.9% unexpected temporary increase

in the nominal rate leads in the case of an accommodative policy, the asset prices to fall by 1 % (both

the market and the fundamental prices), and thus the investment also to fall by 1 %. This pushes

entrepreneurial net worth down by 2 %, resulting in an increase of 1 % in the external finance pre-

mium. This pushes down the level of debt by 3 %. Household consumption is reduced by 0.1 % and

entrepreneurial consumption is also reduced by 1%, all leading the GDP to contract by 0.3 %. Whilst

the general economic contraction is similar for the aggressive policy, the fall in inflation is smaller (0.15

%), and both the policy rate and the inflation return to their steady-state levels sooner. The fall in

asset prices and investment are 0.8 %. The fall in net worth is 1.9 %, leading to a rise in external

finance premium of just under 1 %, and the level of debt to fall by 2.8 %. Household consumption falls

by 0.08 %, leading to a total fall in GDP equal to 0.25 %.

For the last type of shock in this model, a positive disturbance to the market asset price above its fun-

damental value, i.e. the bubble shock in Figure 11, in both policy cases it leads the aggregate demand

to expand and the economy to boom. However, the expansion is more accentuated in the case of an

accommodative policy, since the expansion in firm balance sheet is 20% higher. 8 In the accomodative

case, the market asset price rises by 1.25%, the fundamental price by 0.15% (and subsequently fall 0.05%

in the second period), external finance premium fall by 0.05 %, investment rises by approximately 2.3%,

entrepreneurial consumption increases by 2%, entrepreneurial net worth expands by 2.3 %, leading to

an increase in output of approximately 0.5%. For aggressive monetary policy, on the other hand, the

8These observations are in line with the observations made by Adrian and Shin (2008), and Bean (2008) regarding thehigh procyclical increase in leverage of firms and financial institutions as a response to the increasing asset price bubbleprior to fall of 2007.

12

expansion in aggregate demand is more restricted. The same shock leads the market asset price to rise

by 1.20%, fundamental asset price to rise by 0.05 % (to subsequently fall by 0.1% in the second period),

causing the external finance premium to only fall by 0.035 %, investment to rise by 2.25%, marginal

costs to rise by 0.4 %, entrepreneurial consumption to rise by 2%, and output to rise by 0.5 %. As a

response to the general expansion, the nominal rate rises by 0.8 %, restricting the rise of inflation to

only 0.05 %, instead of the 0.1 % for the accomodative case. But the greatest difference lies of course

in the convergence path of inflation, nominal rate and marginal cost. In all three cases, the return to

the steady state values is smoother for the aggressive policy compared to the accomodative monetary

policy case.

These impulse responses demonstrate that the financial accelerator mechanism amplifies and propagates

the initial shocks. Compared to a standard NK-model, the responses in investment, output and infla-

tion are accentuated, since capital market frictions are propagated to real activity, causing a stronger

cyclical behavior in these variables. In addition, monetary policy does not only influence the economy

through the usual demand channel, but also affects the external finance market by affecting the value

of entrepreneurial collateral as we saw in Figure 1. The policy maker has thus a stronger and more

direct ability to stabilize the economy compared to the standard macroeconomic models which lack the

financial frictions, or which incorporate them in an ad hoc manner.

4.4 Second moment analysis

To asses the impact of alternative monetary policies on the model economy, let us look at the second

moments that the model generates when all shocks are ’switched on’. Table 7 reports the HP-filtered

relative standard deviations and correlations of model key variables with respect to output. In lines

with Bernanke and Gertler (2000), but including more shocks 9 and more model variables, our aim is

to analyze whether an aggressive monetary policy, with (γπ = 2.4), succeeds in brining down a stock

market boom driven by a bubble, and thus stabilize the economy in a better manner than what an

accommodative policy, with (γπ = 1.4), achieves.

From the first two rows of Table 7, we see that for an aggressive policy, the relative standard deviation

of the model variables to output is lower. This means that a more aggressive policy rate manages

in bringing down the cyclical swings in this model economy. Taking further into account that GDP

standard deviation is lower for the (γπ = 2.4)-case, it means that there is some considerable difference in

volatilities between the two cases. The only exceptions are investment, and nominal rate, both slightly

more volatile than the same variables under an accommodative policy. Investment is slightly more

volatile since an aggressive policy manages to depress an investment boom from a bubble, and hence

causes the investment levels to fall slightly more than in an accommodative case. For the policy rate,

the volatility is higher since the rate is more reactive to economic expansions than in the accommodative

case.

The same difference is observed for the correlation coefficients from the last two rows in the same table.

Variables are less correlated to output in the case of an aggressive policy, since such policy manages to

dampen the swings from an economic boom, hence making them less cyclical. The aggressive policy

rate manages to smoothen some of the propagation effects from the financial accelerator mechanism,

and therefore make it less responsive to asset price booms. 10

9Technology, Demand, Monetary policy and Bubble shocks, while in BG (2000), the authors only include Technologyand Bubble shocks

10Just as a minor remark, consumption turns out as slightly countercyclical in the case of an aggressive monetary policysince a relatively higher rel rate reduces consumption of households, thus offsetting the expansionary demand/wealtheffects from a boom.

13

5 Welfare analysis

The previous section computed the impulse response functions and second moments to technology,

demand, monetary policy, and asset-price bubble shocks under alternative monetary policy rules. The

results suggest that there are potential benefits from adopting a policy that implies a strong response

to inflation, in particular for the asset-price bubble shock. Nevertheless, the second moments showed

that the benefits were not very large. This opens up new questions to whether a central bank can

adopt other alternative policy rules to control stock price booms. To further explore these issues, we

will conduct a welfare analysis of alternative monetary policy rules. In particular, we are interested in

maximizing the welfare function for non-asset price targeting, and compare to the welfare when asset

price targeting is included. Our aim is to understand whether a monetary policy that includes stock

market developments achieves better in bringing the economy under control, or whether a sufficiently

and reasonably aggressive inflation targeting is sufficient, as Bernanke and Gertler (2000) showed. Once

that is done, we are then also able to answer our second question on what role the neglecting asset prices

in monetary policy play in the bubble in terms of additional welfare losses?

5.1 Loss function

In the welfare experiments we consider, the government is assumed to minimize a quadratic loss function11 for each monetary rule. In addition to the minimum losses, the optimal weights on each variable

in the monetary policy reaction function are estimated for that specific minimum loss. The following

points summarize the steps we will follow in these experiments:

1. Define the loss function in the certainty equivalence policy experiments. It is important to keep

the loss function constant throughout the experiments.

2. Alternative monetary policy reaction functions are specified. These should reflect alternative

targeting scenarios that the monetary authority is pursuing.

3. The minimum aggregate welfare loss of a particular reaction function is computed. The experi-

ments are repeated for the three different types of bubble processes.

4. The optimal weights of that particular reaction function that generate the minimum loss are

additionally estimated.

5. Alternative reaction functions and their corresponding weights are evaluated using the minimum

loss as a general criterion. The monetary policy that generates the smallest welfare loss for that

specific bubble process will in general be regarded as the most stabilizing one.

The expected loss function the government is assumed to minimize is:

arg minθ∈

`(θj, (α, β, δ)) = Et(4 ∗ αyt + 4 ∗ βπt + 4 ∗ δRt−1) (15)

,with α denoting the weight on output ytin the loss function, β the weight on inflation πt, and δ the

weight on the interest rate lag Rt−1. The corresponding weights of each variable are [1,1,0.25]. θ is the

vector of estimated optimal weights in the monetary policy reaction function that gives the minimum

loss. The function depends on the variance of output, inflation, and the nominal rate. Note that no

11Or welfare function as it is also commonly denoted

14

term appears for the measure of the stock market. This means that in what follows the efficacy of

reacting directly to stock market developments is modeled as the means to an end, and not a goal of

policy itself. The welfare function we use is standard to the literature, and we include a smoothing

variable r in order to avoid excessive volatilities in the policy rate, and hence make it more realistic

(Woodford (2003)).

The above objective function is subject to the rest of the model, the variance-covariance matrix of

stochastic exogenous shocks, and the monetary policy reaction function.

5.2 Monetary policy rules

In order to evaluate the benefits of applying one monetary policy or another, we need to specify the

monetary rules we would like to evaluate. We have already seen that a policy that implies a strong

response to inflation manages to bring an asset price boom under control better than an accommodative

one, as Bernanke and Gertler (2000) pointed out in their experiments. The results they derived are in

an environment where exogenous movements in asset prices provide an additional source of fluctuations

in net worth, but do not alter entrepreneurs’ investment decisions. In our framework, on the other

hand, asset price booms do indeed affect their perceptions regarding the value of new investment, and

so the fluctuations in net worth, investment and external finance premium are more enhanced, as we

have seen. Moreover, we wish to add additional economic settings, and consider the optimal policies to

apply under three alternative asset price bubble situations. Thus, we would like to explore whether their

results still apply under these extensions, or whether other policy targets manage better in stabilizing

the asset price boom, and minimize the welfare losses caused by the bubble.

For this purpose, we will consider five alternative monetary policy rules. The first rule we will consider

is a simple inflation-forecast-based (IFB) rule. IFB rules are preferred to a current-inflation-based rule

for their ability to include a great deal of information in a single object (Coletti (1996)). Levin et al

(2003) find that this type of rules are robust provided that the horizon of the inflation lead is short.The

rule can be written as:

Rt = Et(πt+1) (16)

We use this simple rule as a benchmark case, to which we can compare and contrast the gains of adding

additional variable versus keeping a target simple.

The second policy rule is the one endorsed by Bernanke and Gertler (2000), and which apart from the

IFB rule includes an output target:

Rt = Et(πt+1) + yt (17)

Basing our analysis on the findings made by Cecchetti et al (2000,2003), and Tetlow (2005), a third set

of rules wants to explore whether a monetary policy that allows the nominal rate to respond to stock

market developments can improve upon a policy that responds to inflation only. Since stock market

bubbles are hard to identify, we include three different set of targets which are associated with booms:

asset price bubble, fundamental asset price, and market asset price:

Rt = Et(πt+1) + yt + (∆t − qt) (18)

Rt = Et(πt+1) + yt + qt (19)

15

Rt = Et(πt+1) + yt + ∆t (20)

,where ∆t is the market asset price, qt the fundamental value of assets, and ∆t − qt an asset price

bubble. Taking into account this difficulty, our second aim of this exercise is to find out whether the

two alternative policy target achieves the same level of welfare as the asset price bubble target, but

which is easier to identify. In our experiments, we consider a broader range of stochastic disturbances

compared to BG (2001). Apart from the productivity and bubble shocks, we add a fiscal spending, and

a monetary policy shock. We assume the shocks to be uncorrelated, and to have the variance-covariance

matrix of a diag[1, 1, 1, 1] .

5.3 Type I and II errors

Another issue is related to the information structure of the economy that monetary authorities have.

We have seen previously that the US economy changed structurally around the millennium shift. The

asset price boom during the 2000’s led the financial and real variables to become much more volatile and

cyclical. But such changes are hard to detect at the moment they occur and can lead the central bank

to analyze a model and calibration representing the previous economic structure, while that structure

in the ’true’ economy had changed.12 In other words, the question is what are the losses generated by

the central bank when it thinks there is no asset price boom and does not react to it despite there being

one (Type I error), and when it thinks that there is a bubble and reacts to it, when there is not one

(Type II error)? This can be illustrated by the following diagram:

Table 1: The gamePolicy response/Bubblescenarios

Asset price bubble No bubble

Asset price targeting Relative gains Losses from error type IINo asset price targeting Losses from error type I Relative gains

We are interested in comparing the relative gains from acting against a true bubble versus, not acting,

and compare it to the relative gains from not acting against a bubble versus acting against a bubble,

when there is not one. Hence, first we want to find out the impacts on optimal policy of Type I and II

errors, but more importantly, we would like to find out at what probability, or size of the bubble,13 will

a policy maker be indifferent between targeting a bubble, and not targeting it?

We will do this by calculating the gains from not targeting a market asset price/bubble when there

is no bubble, and search for a level of the parameter ’b’ such that those gains are equal to the gains

from acting correctly to a true bubble. Taking into account that information about the true economy

is imperfect, at that level of the bubble, the central banker will be indifferent between targeting and

reacting to the bubble, as not targeting and reacting to it.

5.4 Bubble processes

Gilchrist and Saito (2006) pointed out in their analysis of the KC-BGG model that their results on

the benefits of reacting to asset price bubbles might be case-sensitive to their calibration of the bubble

process. They conclude that their results might alter if the exogenous asset price movement is calibrated

12Mishkin (2004)13Which is driven by the parameter ’a’

16

in a different manner. Recognizing this case-sensitivity, we extend our welfare analysis and consider

multiple scenarios of the stock market bubble. More specifically, we consider the cases when there is a

mean-reverting temporary bubble that lasts for five periods, a rational bubble in the sense of Blanchard

and Watson (1982), and no bubble. Consistent with Batini and Nelson (2000), the list could be easily

extended to include exchange-rate and commodity-price bubbles, but for the sake of simplicity, we will

focus on the stock market. The key element in the model that determines the bubble process is the

parameter ’b’. It is a bubble parameter, and is fundamentally driven by the expected growth rate

of bubble, and the probability that a bubble persists. Formally, the parameter is composed by the

parameters ’a’, the growth rate of bubble, and ’δ’, the depreciation rate of capital according to:

b = a(1− δ) (21)

The depreciation rate of capital parameter δ is calibrated to a standard value of 0.025, and the parameter

that is varied in the different bubble cases is a. a = 0.98 is the calibration for the mean-reverting

temporary bubble. It lasts for five periods, and since the parameter a is set to less than unity, it

represents the convergence rate to zero of the bubble. The second case, with a = 1 represents a rational

deviation of the price of assets from its present value of expected dividends. This deviation occurs

because either the information on the value of the firm is incomplete, or the market is weakly efficient,

both possible without violating the arbitrage condition (Blanchard and Watson (1982), Blanchard and

Fischer (1989)). The growth rate of the bubble is set at unity, so that investing in the stock market is

profitable. These type of bubbles are only possible if the bubble component of the asset price moves

independently from its’ fundamental component. For the third case, the economy does not display an

asset price boom and so the value of a is set to 1.02564103. Since δ = 0.025, this gives a value of b = 1.

From equations 73 and 74 in the log-linearized model (where the parameter b explicitly enters), the

system of equations are reduced to:

E[r∆t+1] = E[rqt+1] (22)

∆t − qt = (1− δ/Rq)Et [∆t+1 − qt+1] (23)

The first equation implies that the expected return of shares is the same as the expected return from the

fundamental value of assets. The second equation means that the evolution of the bubble is independent

from the ’bubble parameter’, and simply depends on the fraction of net capital over the steady-state

return of the fundamental. Substituting the original equation 74 into 73, and using b = 1, we see that

the right-hand side is canceled, and hence market price equals the fundamental value of assets, i.e. no

bubble.

5.5 Results

Results from the first welfare experiments are reported in Table 8. The column with initial guesses

represent the coefficients/weights given to the different different variables in the reaction function before

the estimation process, while ’Optimal weights’ represents the optimal estimated coefficients/weights on

those variables. Different initial guesses of the monetary policy reaction function parameters/weights

did matter for the minimization of the loss function, and for the estimated optimal parameters, since

csminwel searches for the minimum value of the welfare function around the neighborhood of the initial

17

guesses of the reaction function. We consider the five different monetary policy rules specified above,

and the rule that gives the lowest loss-value is considered to be the one that maximizes the welfare

function.

In the following experiments, we will be looking at mainly three issues. The first question we want to

answer is whether the benefits of responding strongly to inflation exceed the benefits of responding to

stock market developments in terms of welfare maximization. The second issue regards Type I and II

errors, and considering that central bank is uncertain about the ’true’ economy, what are the welfare

costs generated by applying a non-optimal policy? In this instance, we would like to explore whether

Type I and II errors change our choice of optimal policy in the previous case. The third question regards

the economic costs of procyclical monetary policy and asks how much of additional welfare loss does

the economy incur when monetary policy neglects asset prices during a stock market boom?

The first five rows of Table 8 represent the simulations of the mean-reverting bubble, the following five

of the rational bubble in the sense of Blachard and Watson (1982), the subsequent five of the no-bubble

setting, and the final eight represent the control tests that we performed in order to check whether other

weights on inflation, and using forecasted output instead of current output target decreased the losses.

We decided to only consider the aggressive case of inflation targeting, by setting the initial weight on

expected inflation equal to 2, since our results from IRFs, as well as BG(2000) and Gilchrist and Saito

(2006) show that strong response to inflation is strictly preferred. In this way we can concentrate in

evaluating the additional benefits or costs of adding targets on stock market developments. We set the

initial weight on output to 0.5, and in the case of rules including stock market prices, we set the weight

on them equal to 0.2, just as in the BG (2000) paper.

In all three cases of the bubble processes, we see that having the nominal rate respond strongly to

inflation is welfare improving compared to both the case when only inflation is targeted, and when the

fundamental value of asset prices is included in the rule. In percentage terms, having nominal rate

responding strongly to both inflation and output is approximately 35% welfare improving compared

to the other two rules for all bubble scenarios. The reason is that expansionary effects from an asset

price boom are not only captured by inflation, but also output since output is very sensitive to stock

market developments. Hence targeting output provides the policy maker with additional information

regarding the economic activity, and he can more successfully control the economy. For the rule including

the fundamental price of assets, the loss outcome is worse since the model assumption that asset

price bubbles move exogenously means that targeting fundamental value of capital depresses economic

expansions driven by fundamentals, while not controlling the bad effects from asset price booms.

However, the benefits of targeting the asset price bubble, or the market asset price when there is a

bubble are clear. Monetary rules that include one of those two targets are clearly welfare improving.

Compared to an aggressive inflation and output rule only, the increase in welfare by including bubble

or market asset price target is between 19% and 21% in the first two cases of bubbles. Following the

discussion of BG (2001) and Tetlow (2005) that a bubble is hard to identify, a very close substitute to

a bubble-target is to target the market asset price.

Nevertheless, when there is not a bubble, the optimal choice for the policy maker is to aggressively

target inflation and output, but not to include asset price target in any form. Both type of asset price

targets cause an increase in losses, while targeting a bubble is as good as targeting inflation and output-

only since in this case the bubble is equal to zero, and it is therefore the same as targeting inflation

and output-only. What is also striking from our results is that the general level of losses generated

by whatever policy rule for an economy without a bubble are higher than when a bubble exists. This

means that any type of policy rule that responds strongly to economic/market conditions would in the

case of an economy driven by fundamentals cause the growth to be reduced, and cause higher losses.

18

As a control exercise, we also performed some additional simulations of the model, where we increased

the responsiveness of nominal rate to expected inflation to 3, and in a second step, changed all the

reaction functions to include an forecast-output-based rule instead of the current output. For the first

modification, we see that for the mean-reverting-bubble case, all rules generate a smaller loss, meaning

that increasing the responsiveness to expected inflation improves welfare. For an expected inflation and

output-only rule, the welfare improvement is around 10%, and for a rule that includes the bubble, the

improvement is 4%. This means that, in lines with BG (2000) and Tetlow (2005), that a sufficiently

strong inflation target might do as good job, or even better than a stock market price, or even a bubble

target. We will explore this issue in the next section when we look more closely at the shape of the loss

functions. For the second control test, the losses generated are higher, which means that a rule that

includes a current output target performs better in welfare terms.

Answering our second question regarding errors type I and II, we see that not targeting the bubble when

there is one is clearly welfare reducing in both bubble scenarios, since the increase in loss is 27% in the

case of a mean-reverting bubble, and 28% in the case of a rational one. For error type II, on the other

hand, targeting the asset price when there is no bubble is welfare reducing since the strong response

of the nominal rate to asset prices despresses ’healthy’ economic growth, increasing thus the economic

losses. We will look at this in more detail in the next section when we conduct certainty equivalence

experiments.

For the third question, we see that the economy incurs an additional 27% of losses in the case of mean-

reverting bubble, and 28% of losses in the case of a rational bubble, when the central bank neglects

asset prices in their rule, and there is a boom on stock markets. However, the welfare loss is reduced

by 37% for the mean-reverting bubble scenario, and by 39% for the rational bubble scenario when the

monetary authority neglects fundamental asset prices in their target, compared to when they do not.

Hence, it is important that the central bank responds to the exogenous movements in asset prices, and

not to the endogenous fundamental movements if it wants to maximize welfare.

5.6 Extended analysis of the loss function

Recognizing the results of BG (2000) that a sufficiently strong response of nominal rate to expected

inflation is more beneficial than responding to asset prices, we want to extend our range of experiments

for the bubble setting and compare whether a rule that reacts sufficiently strong to expected inflation

but does not include stock market developments can achieve as good welfare maximization as a rule that

includes a stock market price term. More importantly, this will assist us in understanding at what stage

that indifference-of-policies-point occurs, i.e. a probability/size of the bubble which makes the policy

maker indifferent between targeting asset prices and not targeting them. We will try to endogenously

search for that point.

At the same time, we will be able to depict the loss function more carefully, and understand whether

the minimum losses for each reaction function and bubble scenario that we obtained in the previous

section is a local, or a global minimum, and the gradient of the function around that global minimum.

We simulate our loss function for our five different monetary policy rules, and for each bubble scenario.

Tables 9 to 14 report the results from our experiments. For each policy rule, and bubble process, we

conducted multiple simulations where we varied the initial weights of the variables in the reaction func-

tion, and observed the loss value. We changed only one initial weight/parameter at a time so that we

could appreciate the gradient of the loss function.

Let us start by tackling our second issue of depicting the loss function. For this exercise, we performed

approximately 20 simulations per reaction function, and per bubble process.

19

For an economy with a mean-reverting bubble, we see that the inflation-only policy loss function is the

flattest, with two local minima around the estimated optimal weights of 0.94 and 44.03. The minimum

loss in both cases is 0.0051, and when the initial weights were modified, the estimated losses, and the

optimal weights all converged to those values.

The loss function simulated for an asset-price bubble rule has a more elastic loss function than the

former, but still represents the second flattest. Its global minimum occurs in the interval of optimal

vector of weights [1.81;0.63;0.33] and [5.12;0.78;0.38] with the estimated loss of 0.0024.

The subsequent three loss functions in order of ascending steepness are functions simulated for a market-

asset price,- output-and expected inflation-only,- and fundamental-asset price rules. While the loss

function of the policy rule including market asset prices has a global minimum at 0.0024, the latter two

have multiple local minima, but the loss function never reaches the lower bound of 0.0024 for whatever

weights on the variables in the reaction function. This means that for a mean-reverting bubble, only

the policy rules that include asset price-bubble,- or market-asset price target can achieve the maximum

welfare in terms of losses for a combination of joint shocks to the economy.

For a rational bubble, the situation is very similar. The loss function that simulated for an inflation-only

policy is the flattest, with two local minima of 0.0049 for the optimal weights of 0.94 and 44.03. The

next two in the lead are loss functions simulated for policies including a bubble target, or a market-

asset price target, in order of ascending gradient. They succeed in reaching the global minimum loss

of 0.0024. While the estimated optimal vector of weights for the bubble targeting is [2.63;0.68;0.38],

the market-asset price policy achieves it with a multiple pair of estimated optimal vector of coefficients:

[1.63;0.88;0.39]; [1.78;0.82;0.42]; [1.82;1.18;0.38]; [1.91;2.09;0.29]; [2.22;0.78;0.48]; and [2.68;0.82;0.52].

For both rules, they include a strong response of nominal rate to expected inflation. Loss functions of

inflation-and output-only policy succeeds in approaching the global minimum, hitting as low as 0.0026,

but it requires a very aggressive vector of optimal weights on expected inflation and output to reach

that outcome: [2.69;2.81], or above. Just as before, loss functions for rules including fundamental asset

prices, with the steepest loss function, does not manage to reach the same low loss as the previous ones.

Finally, for the no-bubble economy, the ranking in steepness of the loss functions is the same as in

previous cases, but the loss functions for the policy rules that include asset price bubble, market asset

price, and exclude any asset price target have very similar steepness, so similar that they are very hard

to distinguish. Only the loss function simulated for an expected inflation-and output-only rule achieves

the smallest loss of 0.0067 in this economy.

Turning now to our first issue, we would also like to explore what optimal vector of weights are required

for the numerous monetary policies and bubble scenarios to achieve the global minimum loss. From

our experiments, we find that for an economy with the parameter b = 0.98 on the bubble, a policy rule

including a bubble target achieves this minimum when a slightly higher weight on inflation is used, i.e.

[2.06;0.64;0.34]; or [2.81;0.69;0.39], compared to the rule including market-asset prices, where a higher

weight is put on output, as in [1.83;1.17;0.37]; and [1.91;2.09;0.29].

For the rational bubble case, again a policy rule including a bubble target requires a higher weight on in-

flation and the bubble term to achieve the global minimum loss with the estimated vector [2.63;0.68;0.38],

whilst the market-asset price policy requires a higher weight on output, with the optimal vector of

[1.91;2.09;0.29]. For the loss function of the expected inflation and output-only rule to approach the

global minimum, a high weight on inflation and output is required in the policy target of between

[2.69;2.84] and [5.89;4.61].

20

5.6.1 When is the policy maker indifferent between targeting and not targeting asset

prices?

Let us now turn to our main issue in this exercise. Since targeting asset prices when there is a bubble

is optimal, while not targeting it when there is not a bubble is the best option, we need to analyze the

break point, and understand when a policy maker will be indifferent between applying the two policies.

We know that beyond that point, as the probability of a bubble grows, the gains from targeting a true

bubble increase, at the same time as the losses from error type II also increase, and vice versa for the

probabilities below that point. Hence, the impact of incomplete information on the choice of optimal

policy will be equally distributed between the two error types.

From the initial experiments in Table 8, we see that the gains from targeting a true bubble in welfare

terms is 0.0007 for the mean-reverting, and 0.0006 for the rational bubbles. On the other hand, the

gains from not targeting asset prices when there is not a bubble is 0.0001. However further experiments

from Tables 9 to 14 show a more detailed picture. In particular, when the bubble is rational (a = 1),

we see that the gains from targeting asset prices is also equal to 0.0001 in cases where the policy

maker applies a very aggressive inflation and output rule with optimal weights equal to [2.69;2.84] and

[5.89;4.61] for error type II. Recognizing that these cases are very restrictive and require a very strong

response in both expected inflation and output, we want to extend our experiments, considering a case

when the probability or size of a bubble is very low. In particular, we conducted the same welfare

experiments as above for a = 1.025. Keeping δ constant and equal to 0.975, this would mean that

the bubble lasts for maximum one period. The results are reported in Table 15. Under this type of

bubble process, more reasonable weights on expected inflation and output are required to generate a

gain from targeting asset prices when there is a bubble equal to 0.0001. We see that optimal weights

such as [1.88;2.13] and [2.14;2.36] are required for the non-asset price policy to generate a loss of 0.0026.

At the same time, a wide range of optimal weights in the market-asset price rule generate a loss of

0.0025, such as [1.69;0.81;0.41], or [1.62;0.88;0.39], or [2;0.50;0.50]. For those combinations of optimal

weights, amongst others, the gains from targeting a true bubble would be 0.0001, which is equal to

the gain of not responding to asset prices when there is no bubble. Hence, at a = 1.025, the policy

maker will be indifferent between these two rules, and since the size of the bubble is small, the risks

from not identifying correctly a bubble cancel the gains from responding correctly to a bubble, and so

a policy maker can afford to neglect stock prices from its targeting.For higher probabilities or sizes of

the bubble, this will not be true.

6 Conclusion

The paper has analyzed two things. The role that a procyclical monetary policy play in the stock

market boom, as well as the welfare gains from applying an optimal policy, i.e. how much less of a

bubble there would have been if a different monetary policy had been employed. We used an extension

of the financial accelerator model by Bernanke-Gertler-Gilchrist (1999) to understand how stock market

bubbles intensify the business cycles in an economy with financial market imperfections. Our efforts

were concentrated in three directions. First, we drew some stylized facts about the US economy since

the post-war period, and examined the impact that the asset price boom of 2000’s have had on US

financial and economic variables. We find that both the corporate and household variables had become

much more volatile and procyclical since the millennium shift. Moreover, we find that financial spreads

had become more countercyclical during the past ten years, while the funds rate had become nearly fully

procyclical. Hence, we conclude that the asset price boom had structurally changed the US economy,

21

making it much more volatile and cyclical.

Second, we embellished the KC model adding a stronger behavioral link between investment and the

stock market to study the contribution from procyclical monetary policy to asset price boom. Third,

we conducted a multitude of welfare experiments, and examined the welfare maximizations generated

by alternative monetary policy rules. Our findings suggest that a policy maker who fails to recognize a

stock market boom, and neglects asset prices from his policy target contributes to greater welfare losses

from the stock market boom. The results from the experiments show that for whatever definition of

the bubble process, a policy rule that targets an asset price bubble, or market asset prices is always

preferred to a policy that neglects it. In the context of the model, we interpret the failure to target the

bubble as welfare reducing since the policy maker will fail to control all of the demand-expansionary

effects from a stock market boom, and thus contribute to greater economic instability.

This paper focuses on an approximated standard interest-smoothing loss function rather than a formal

welfare function derived from first principles. While the version of the welfare function we use is

standard in the literature, and a good estimation of the losses for the type of models that we use

here, it is nevertheless an approximation, and not an exact one for our specific model. Thus, future

work should be oriented towards assessing the robustness of our conclusions for welfare calculations.

In addition, although the BGG model incorporates financial imperfections, it would be interesting to

assess the economic impacts of incorporating an endogenous intermediation sector as a driver of market

liquidity for magnitudes that we observed on markets during the 2000’s. we are therefore also interested

in exploring the robustness of our conclusions to these alternative mechanisms that may provide a more

realistic characterization of the financial sector.

22

References

Allen, Franklin and Douglas Gale, (2001), ”Bubble and Crises”, The Economic Journal (Dec)

Batini, Nicoletta and Edward Nelson, (2000), ”When the Bubble Bursts: Monetary Policy

Rules and Foreign Exchange Market Behavior”, unpublished manuscript, Bank of England

Bernanke, Ben and Mark Gertler, (1999), ”Monetary Policy and Asset Price Volatility” Fed-

eral Reserve Bank of Kansas City Economic Review, Fourth Quarter 1999, 17-51

Bernanke, Ben and Mark Gertler, (2001), ”Should Central Banks Respond to Movements in

Asset Prices?”, American Economic Review 91 (2), 253-257

Bernanke, Ben, Mark Gertler and Simon Gilchrist, (1999), ”The Financial Accelerator Model

in a Quantitative Business Cycle Framework”, in john B. Taylor and Michael Woodford (eds.),

Handbook of Macroeconomics, Vol 1-C, Amsterdam: Elsevier Science, North Holland

Blanchard, Oliver Jean and Mark Watson, (1982), ”Bubbles, Rational Expectations and Fi-

nancial Markets”, in P. Wachtel (ed.), Crisis in the Economic and Financial Structure, Lexington

Books

Blanchard, Oliver Jean and Stanley Fischer, (1989), Lectures on Macroeconomics, Cambridge

MA:MIT Press

bond, Stephen and Jason Cummins, (2001), ”Noisy Share Prices and the Q Model of Invest-

ment” Institute for Fiscal Studies Economic Review 22, (Winter)

Bullard, James B. and Eric Schaling, (2002), ”Why the Fed Should Ignore the Stock Mar-

ket”, Federal Reserve Bank of St. Louis Review

Calvo, Guillermo, (1983) ”Staggered Prices in a Utility-Maximizing framework”, Journal of

Monetary Economics 12, 383-398

Chadha, Jagjit, Germana Corrado and Luisa Corrado, ”Consumption Dynamics: Housing Col-

lateral and Stabilization Policy”, Paper presented in honor of Stephen Turnovsky in Vienna, May 2010

Checcetti, Stephen G., Hans Genberg, John Lipskey and Sushil Wadhwani, (2000), Asset

Prices and Central Bank Policy, Geneva Report on the World Economy 2, Geneva: International

Center for Monetary and Banking Studies and Center for Economic policy Research

Checcetti, Stephen G., Hans Genberg and Sushil Wadhwani, (2003), ”Asset Prices in a Flexi-

ble Inflation-Targeting Framework” in Hunter et al, 427-444

Clarida Richard, Jordi Gali and Mark Gertler, (1998), ”Monetary Policy Rules in Practice:

Some International Evidence” European Economic Review (June), 1033-1068

23

Coletti, Donald, Benjamin Hunt, David Rose and Robert J. Tetlow, (1996), ”The Bank of

Canada’s New Quarterly Projection Model, part 3, The Dynamic Model: QPM”, Bank of Canada

Technical Report (75), Bank of Canada, Ottawa

Dupor, Bill (2005), ”Stabilizing Non-Fundamental Asset Price Movements under Discretion

and Limited Information”, Journal of Monetary Economics 52, 727-747

Gali, Jordi, (2008), Monetary Policy, Inflation and the Business Cycle: An Introduction to

the New-Keynesian Framework, Princeton and Oxford:Princeton University Press

Giannoni, Marc P., (2002), ”Does Model Uncertainty Justify Caution?: Robust Optimal Pol-

icy in a Forward-looking Model”, Macroeconomic Dynamics (February)

Gilchrist, Simon and John V. Leahy, (2002), ”Monetary Policy and Asset Prices”, Journal of

Monetary Economics 49, 75-97

Gilchrist, Simon and Masashi Saito, (2008), ”Expectations, Asset Prices and Monetary Pol-

icy: The Role of Learning”, in John V. Campbell (ed.) Asset Prices and Monetary Policy,

Chicago:University of Chicago Press

Hall, Simon (2001), ”Credit Channel Effects in the monetary Transmission Mechanism”, Bank of

England Quarterly Bulletin (Winter)

Hunter, William C., George G. Kaufman and Michael Pomerleano (eds.), Asset Price Bub-

bles: The Implications for Monetary, Regulatory and International Policies, Cambridge MA:MIT

press

Kiyotaki, Nobuhiro and John Moore, (1997), ”Credit Cycles”, Journal of Political Economy

105, 2 (Apr), 211-48

Levin, Andrew, Volker Wieland and John C. Williams, (2003), ”The Performance of Forecast-

based Monetary Policy Rules Under Model Uncertainty”’, American Economic Review 3 (June),

622-645

Ludvigson, Sydney C. and Charles Steindel, (1999), ”How Important is the Stock Market Ef-

fect on consumption?”, Federal Reserve Bank of New York, Economic Policy Review (July)

Modigliani, Franco and Merton miller, (1958), ”The Cost of Capital, Corporate Finance and

the Theory of Investment”, American Economic Review 48 (June), 261-97

Mussa, Michael, (2002), ”Asset Prices and Monetary Policy”, in Hunter et al, 41-50

Sims, Christopher A. and Zha Tao, (2006), ”Were There Regimes Switches in US monetary

policy?”, American Economic Review 96, 1, (March), 54-81

Stock, James H. and Mark W. Watson, (1998), ”Business Cycle Fluctuations in US Macroe-

24

conomic Time Series”, in J. B. Taylor and M. Woodford (ed.), Handbook of Macroeconomics, edition

1, volume 1, 3-64, Elsevier.

Tetlow, Robert J., (2005), ”Monetary Policy, Asset Prices and Misspecification: The Robust

Approach to Bubbles with Model Uncertainty”, Manuscript, Board of Governors of the Federal

Reserve System

Woodford, Michael, (2003), ”Optimal Interest-rate Smoothing”, Oxford University Press and

The Review of Economic Studies, 70, 4, (Oct), 861-886

Woodford, Michael, (2003), Interest and Prices: Foundation of a Theory of Monetary Policy,

Princeton, NJ:Princeton University Press

25

Appendices

A Annex A

A.1 The model

A.1.1 The optimization problems

The representative risk-averse household maximizes its lifetime utility, which depends on consumption

Ct+k, real money balances M/P t+k and labor hours (fraction of hours dedicated to work=Ht+k):

maxEt

∞∑k=0

βk[ln(Ct+k)− ςlnMt+k

Pt+k+ θln(1−Ht+k)] (24)

constrained by lump-sun taxes he pays in each period Tt, wage income Wt, and dividends he earns in each

period from owning the representative retail firm∏

t, and real savings he deposits in the intermediary,Dt

according to the budget constraint:

Ct = WtHt − Tt +∏t

+RtDt −Dt+1 + b[Mt+1 −Mt

Pt](25)

The household takes Wt, Tt, and Rt as given and chooses Ct, Dt+1, Ht and M/P t to maximize its utility

function subject to the budget constraint.

The other key agent in this model, the representative entrepreneur, chooses capital Kt, labor input

Lt and level of borrowings Bt+1, which he determines at the beginning of each period and before the

stock market return has been determined, and pays it back at the end of each period as stated by

Rt+1[QtKt+1 − Nt+1] (where Rt+1 is the risk-free real rate that borrowers promise lenders to pay back

on their loans, ∆t is the market price of equity at ’t’, Kt+1 is the quantity of capital purchased at ’t+1’

and Nt+1 is entrepreneurial net wealth/internal funds at ’t+1’) to maximize his profits according to:

V = maxEt

∞∑k=0

[(1− µ)

∫ $

0

ωdFωU rkt+1]Et(R

qt+1)QtKt+1 −Rt+1[QtKt+1 −Nt+1] (26)

with µ representing the proportion of the realized gross payoff to entrepreneurs’ capital going to mon-

itoring, ω is an idiosyncratic disturbance to entrepreneurs’ return (and $ is hence the threshold value

of the shock), EtRqt+1 is the expected stochastic return to stocks, and U rk

t+1 is the ratio of the realized

returns to stocks to the expected return (≡ Rqt+1/EtR

qt+1). The entrepreneur uses household labor

and purchased capital at the beginning of each period to produce output on the intermediate market

according to the standard Cobb-Douglas production function:

Yt = ztKαt L

1−αt (27)

where Yt is the output produced in period ’t’, zt is an exogenous technology parameter, Kαt is the

share of capital used in the production of output, and L1−αt is the labor share. The physical capital

accumulates according to the law of motion:

Kt+1 = ΦItKt

Kt + (1− δ)Kt (28)

26

with Φ ItKtKt denoting the gross output of new capital goods obtained from investment It, under the

assumption of increasing marginal adjustment costs, which we capture by the increasing and concave

function Φ. δ is the depreciation rate of capital. The entrepreneur borrows funds from the financial

intermediary, as a complement to its internal funds, in order to finance its purchase of new capital,

which is described by the following collateral constraint:

Bt+1 ≤ Nt+1 (29)

and the cost of external funding, which is represented by the external finance premium (EFP) condition:

Et(Rkt+1)

Rt+1

= s[Nt+1

QtKt+1

] (30)

By assuming a fixed survival rate of entrepreneurs in each period, the model assures that entrepreneurs

will always depend on external finances for their capital purchases, and are further assumed to borrow

the maximum amount, subject to the value of their collateral (which means that the collateral constraint

will bind with equality).

To complete the model, let us look at the maximization problem of the remaining agents in the model:

financial intermediaries, retailers and government. The role of the financial intermediary in this model

is to collect the deposits of savers, and to lend these funds out to borrowers through 1-period lending

contracts against a risk-free return Rt that households demand on their deposits. Because of information

asymmetries between lenders and borrowers, the intermediary needs to invest some costly monitoring

of the borrowers in order to assure that borrowers survive to the next period and pay back the return

on deposits. Most importantly, the intermediary can not lend out more than the deposits they have

(incentive constraint), and it operates in a perfectly competitive market. This means that in each

period, intermediaries choose a level of borrowings Bt in order to maximize:

maxF = Et

∞∑t=0

(Rt−1Bt −Bt+1)− (Rt−1Dt −Dt+1)− µ(ωRt+1QtKt+1) = πt (31)

where µωRt+1QtKt+1 is the monitoring cost of borrowers. The amount of lending is constrained by the

incentive constraint:

Bt+1 ≤ Dt+1 (32)

and the intermediary makes zero profits in each period:

∞∏t=0

πt = 0 (33)

To incorporate the nominal rigidities, a standard feature of New-Keynsian models, we incorporate a

retail sector into this model. Let us look at retailers’ problem. They set their price of the final good

according to the standard Calvo process (1983), where P ∗t is the price set by retailers who are able to

change prices in period ’t’, and let Y ∗t (z) denote the demand given this price. Then, retailer ’z’ chooses

P ∗t in order to maximize:

maxR = θkEt−1

∞∑k=0

[Λt,kP ∗t − P ω

t

PtY ∗t+k(z)] (34)

27

with θk being the probability that a retailer does not change his price in a given period, Λt,k ≡ βCt/Ct+1

denoting the household intertemporal marginal rate of substitution (since households are the sharehold-

ers of the retail firms), which they take as given, and P ωt ≡ Pt/Xt denoting the nominal price of goods

produced by a retailer (Xt is the gross markup of retail goods over wholesale goods). They face a

demand curve equal to:

Yt(z) = (Pt(z)

Pt)−εY f

t (35)

where Pt(z)/Pt−ε is the nominal price ratio of wholesale goods for retailer ’z’, ε > 1 is a parameter on

retail goods, Y ft is the total final output in the economy which is composed by a continuum of individual

retail goods, and Pt is the composite nominal price index of a continuum of individual prices set by

retailers. Finally, a government plans spending, and finances it by either lump-sum taxes, or money

creation (Central Bank division). In each period, it chooses spending Gt, and a combination of taxes

Tt and money creation Mt so to fulfill the balanced budget condition:

Gt =Mt −Mt−1

Pt+ Tt (36)

It chooses money creation for budget financing according to a standard Taylor rule:

Rnt

Rnt−1

= ρ+ ξπt−1 (37)

where Rnt is the policy rate in period ’t’, ρ is the coefficient of interest rate growth, and ξ is the coefficient

on lagged inflation in the Taylor rule.

A.1.2 Solutions

Solving the household’s optimization problem yields standard first order conditions for consumption,

1

Ct= Et(β

1

Ct+1

)Rt+1 (38)

labor supply,

Wt1

Ct= θ

1

1−Ht

(39)

and real money holdings,Mt

Pt= ςCt[

Rnt+1 − 1

Rnt+1

]−1 (40)

The last equation implies that real money balances are positively related to consumption, and negatively

related to the nominal/policy interest rate. Turning to the entrepreneur, his choice of labor demand,

capital, and level of borrowings yields the following optimization conditions:

(1− α)YtLt

= XtWt (41)

(42)

(43)

28

As is common in the literature, marginal product of labor equals their wage. Labor input in the

production of wholesale goods can either come from household or entrepreneur’s own labor supply (ie.

they can devote a small fraction of their own time to the production activity). Therefore entrepreneurs

receive income from supplying labor based on the wage rate above. Since that income stream is assumed

to be marginal in this model however, we can assume that the proportion of entrepreneurial labor used

for production of wholesale goods is so low that it can be ignored, so that all of labor supply is provided

by the household sector.

Continuing our analysis with the retailers, and differentiating their objective functions with respect to

P ∗t gives us the following optimal price rule:

θkEt−1

∞∑k=0

[Λt,k(P∗t /P

∗t+k)

−εY ∗t+k(z)[P ∗t − (ε

ε− 1)P ω

t ]] = 0 (44)

Retailer’s price will be set such that in expectation terms discounted marginal revenue equals discounted

marginal cost, subject to the Calvo price setting probability θ. Since only a fraction of retailers will set

their price in each period, the aggregate price evolves according to:

Pt = [θP 1−εt−1 + (1− θ)(P ∗t )(1− ε)] 1

1− ε(45)

To conclude the optimizations, we turn to the representative intermediary. He will set the maximum

level of lending such that the incentive constraint is satisfied, and subject to the competitive market

condition. Differentiating his value function with respect to Bt+1 will mean that the level of credit given

to the entrepreneurial sector will be:

Bt+1 = Dt+1 (46)

This condition will hold in each period, which means that the intermediary’s balance sheet expansion

is limited to its deposit holdings in each period.

A.1.3 Log-Linearization of the original model

With the optimization problems defined, and the first order conditions obtained, we can now specify

the system of equations constituting the general equilibrium.

Combining the spending of the five agents in this economy, we get a relatively standard Aggregate

Demand equation, modified to accommodate for entrepreneurial consumption, and spending on mon-

itoring, which in the log-linearized version is assumed to be small and of second order. Hence, the

resource constraint consists of a weighted sum of household consumption, entrepreneurial consumption,

investment and government spending. Household consumption is represented by the Euler equation

from household’s first order condition wrt. consumption. Entrepreneurial consumption is assumed

to be dependent on investment demand, so that whatever proportion of entrepreneurial net wealth

that is not re-invested, will be consumed Cet = 1 − It. In the log-linearized model, the proportion of

entrepreneurial consumption contributing to the total demand is kept small. Further, because of the

endogenous collateral constraint introduced in this framework, investment is defined by three equations.

Recall that the entrepreneur holds limited internal funds, and needs to borrow externally in order to

fully finance his investment demand. Because of increasing default risks with increasing leverage, he is

allowed to borrow as much as his collateral permits him, i.e. the EFP constraint in his optimization

problem (which is derived from intermediary’s optimization problem and entrepreneur’s demand for

29

capital). Hence, the amount he can invest will directly depend on the supply of investment finances,

which is the first expression defining his investment expenditure in the log-linearized model. Differenti-

ating entrepreneur’s maximization problem with respect to capital gives us the second determinant of

the investment, which defines his demand for new capital. As we will see in the log-linearized equations,

the return on capital depends inversely on the level of investment, reflecting diminishing returns. The

last equation, that is directly derived from the law of motion of capital in entrepreneur’s optimal choice

problem, defines the determinants of share prices. It establishes the positive relation between the price

of shares and investment demand. Thus a higher market price of capital will stimulate the investment

demand and relax the borrowing constraint that the entrepreneur faces, increasing the supply of external

finances, resulting in an increase in investment expenditure, which leads to an increase in output, and

an increase in the market price of capital. The final component of the aggregate demand, government

spending, is determined by the government sector and satisfies a simple public budget constraint, but

because it is of second order importance for the stability of the general equilibrium, we do not specify

it in the log-linearized model.

The supply side is relatively conventional in this model. The aggregate production is defined by Cobb-

Douglas production function. Recalling from the previous section, since the entrepreneurial sector only

provides a very small fraction of the total labor supply, we assume that the labor input that matters for

the production of wholesale goods is the one from households, and therefore only include them in the

log-linearized equations. Next, we characterize the labor market equilibrium, which is the log-linearized

version of the demand for household labor, weighted by the marginal utility of consumption that, be-

cause of logarithmic preferences of households, is equal to −ct. We continue by defining the Phillips

curve. We obtain it by combining the aggregate price index of retailers with the first order condition

of the same and log-linearize it. In the final expression, the markup xt varies inversely with demand,

since a rise in sales as a result of a rising demand for the retailers that can not change their price at

time ’t’ means that their purchase of wholesale goods from the entrepreneur will increase, pushing up

the relative price of wholesale goods, and inversely pressing down the markup.

The final section of the log-linearized model describes the evolution of state variables, the monetary

policy and the shock processes. The evolution of capital is obtained by log-linearizing the law of motion

constraint in entrepreneur’s optimization problem, once the adjustment cost function has been normal-

ized so that the price of shares is unity in the steady state. Net wealth accumulation is obtained by

log-linearizing the final expression for entrepreneur’s net wealth, which includes the equity from capital

investments, and income from supplying labor:

Nt+1 = γ[RktQt+1Kt − (Rt

µ∫ $

0ωdFωRk

tQt+1Kt

Qt+1Kt −Nt−1

)(Qt+1Kt −Nt−1)] + (1− α)(1− Ω)ztKαt H

(1−α)Ωt (47)

where γ is the survival rate of entrepreneurs, and H1−αΩt is the labor input in the aggregate production

coming from entrepreneurs. We assume that last expression to be fixed to unity so that the portion of

income coming form supplying labor is marginal, and can be expressed as a second order term. The

uncertainty component of the monitoring cost expression∫ $

0ωdFω will be captured by the leverage

ratio in the log-linearized version χ(KN

) since the impact of uncertainty on the cost of monitoring will

positively depend on how much loans the entrepreneur has taken out in relation to his internal funds, i.e.

the leverage ratio. Monetary policy is given by a conventional Taylor rule adapted to include inflation

expectations in the economy. Lastly to complete the model, we specify the four exogenous shocks. We

consider here shocks to asset price εq, to the nominal rate εrn , to fiscal policy εg, and to productivity

zt. They follow a first-order vector autoregressive stochastic process.

30

A.2 The Log-Linearized equations

The competitive equilibrium is characterized by a set of prices and quantities, such that:

Given Wt, Rt, ξ, θ,Dt the household optimizes Ct, Ht, Dt+1

Given Wt, Rt, Rqt , γ, δ, zt, µ,Nt, Kt the entrepreneur optimizes It, Ht, Kt+1, Bt+1, Nt+1

Given Rt, Rqt , Dt+1, Bt the financial intermediary optimizes Bt+1

Given Λt,k, θ, Yt(z), P ωt the retailer ’z’ optimizes P ∗t

Labor, capital and financial markets clear: Hst = Hd

t , Kst = Kd

t , Dt = Bt

,and

Aggregate demand holds: Yt = Ct + It + Cet +Gt

The complete log-linearized BGG framework is presented below by the Equations 48 through 66. In

all the equations, lower case letters denote percentage deviations from steady state, and capital letters

denote steady state values:

The main body of the model:

yt =C

Yct +

Ce

Ycet +

I

Yit +

G

Ygt + εgt (48)

ct = −σrnt+1 + E[ct+1] (49)

cet =K

Nrkt − (

K

N(1− N

K)ψ)rt−1 − (

K

N(1− N

K)ψ)kt−1 −

(K

N(1− N

K)ψ)qt−1 + (

K

N((1− N

K)ψ) +

N

K)nt−1 (50)

Et[rqt+1] = (1− ε)(xt + yt − kt−1) + εqt − qt−1 (51)

Et[rqt+1]− rt = −ψ(nt − qt − kt) (52)

st = Et[rqt+1]− Etrt+1 (53)

Et(qt+1) = φ(it+1 − kt+1) (54)

yt = zt + αkt−1 + (1− α)lt (55)

yt + xt −1

σct = γlht (56)

31

Et[πt+1] = κEt[xt+1] + γfEt[πt+2] + γbπt (57)

nt = χrkt − χ(1− N

K)rt−1 − χ(1− N

K)ψkt−1 − χ(1− N

K)ψqt−1 +

χ(1− N

K)ψ +

N

K)nt−1 + (χ(1− γrkss) +

N

K)/γ)yt (58)

kt = δit + (1− δ)kt−1 (59)

levt+1 = (qt + kt+1)− nt (60)

rnt = ρnrnt−1 + ρππt + ρyyt + εi,t (61)

rt = rnt − Et[πt+1] (62)

Shock processes:

εrn

t = ρrn + urn

t (63)

εgt = ρgεgt−1 + ugt (64)

zt = ρzzt−1 + εz (65)

εqt = ρεuqt (66)

A.3 Extension of the canonical linear model

To accommodate for an exogenous bubble in asset prices, we slightly modified the canonical BGG

model. In particular, we added a parameter on the evolution of the bubble component b 14 to the

baseline model, as well as two endogenous (price of shares, and return on shares in period ’t’) and two

predetermined (price of shares, and return on shares variables in ’t-1’)) variables, following KC (2000).

Three equations were added and six baseline equations were modified. The modified equations are the

’investment incentive’ (the relation between price of external fund and interest rate) that is changed to

E[r∆t+1]− rt = −ψ(nt −∆t − kt+1) (67)

’asset price’ equation is modified to,

Et∆t+1 = φ(it+1 − kt+1) (68)

14which depends on the rate of convergence of the bubble a, and the probability the bubble persists in a given period p

32

Table 2: Model variablesSymbol Variabley Outputc Household consumptionce Entrepreneurial consumptioniv Entrepreneurial investmentg Government spendingx Marginal costl Hours of labor inputn Net worth of entrepreneursk Capitals External finance premiumlev Level of debt of entrepreneursq Fundamental asset pricerq Return on market price of assets/Loan rater∆ Return on fundamental price of assetsrn Nominal interest rater Risk free return on savingsπ Inflation

’the external finance premium’ equation is modified to,

st = E[r∆t+1]− Ert+1 (69)

’consumption of entrepreneurs’

cet = ∆t +Kt+1 (70)

’equity accumulation’ that is changed to

nt = χ[K/Nr∆t − Et−1r

∆t + (1− γRKss)/γ)yt + nt−1] (71)

and ’the Phillips curve’ that in the KC-BGG version is

Et−1[πt] = κE[xt] + γfEt[πt+1] + γbπt−1 (72)

The three equations that were added are the ’expected evolution of the bubble’

∆t − qt = 1− δ/bRqEt∆t+1 − qt+1 (73)

’relationship between stock and fundamental return’

E[r∆t+1]− E[rqt+1] = −(1− b)(∆t − qt) (74)

’return on shares’

E[r∆t+1] = (1− ε)(x+ y − kt−1) + ε∆t −∆t−1 (75)

33

B Welfare function estimation strategy

To estimate the loss function for the different policy options, we make use of Sims codes, which represent

a robust computational method for rational expectations models based on the QZ matrix decomposi-

tion. The program solves nonlinear equations, and minimizes them using quasi-Newton methods with

BFGS updates. The particular strength of these codes lies in the ability of the minimizer to negoti-

ate hyperplane discontinuities without getting stuck, allowing us to to estimate our loss function more

smoothly and efficiently. The only drawback that has been noted until now is that, despite it being

more robust against some specifications of the likelihood function, in some cases it is not very clear

whether it succeeds in jumping the ’cliffs’ in a reliable way, and hence requires a broader depiction of

the loss function around the estimated minimum.15

The key file that we call on in our estimations is the csminwel.m file that draws on our loss function

defined in lsscby.m, and estimates it using the inverse Hessian, that has to be positive definite in

order for a solution to exist. We therefore need to provide some initial values of the inverse Hessian

in csminwel.m which will ensure that it is positive definite. Alongside we need to provide the initial

(non-estimated) values of the parameter vector in our monetary policy reaction function that we want

to estimate and a convergence criterion. 16 One can also specify the maximum number of iterations

for one simulation, but if the critical threshold above was specified, then that is not necessary. Finally,

one is free to either specify a string of the function that calculates the gradient of the loss function, or

if left in blank, the program will calculate a numerical gradient (which is our case). Since we are also

estimating the optimal vector of parameters that minimizes the loss function, we need to provide an ex-

plicit policy reaction function in lsscby.m, which we define based on the simulated DSGE model. With

these inputs, the csminwel.m routine calls on csminit.m that searches for a loss function minimum

around the neighborhood of the initial vector of parameters input in csminwel.m, and estimates the

corresponding vector of parameters in the reaction function that produces that minimum loss. Hence,

different initial values of the vector of reaction function parameters will give different minimum losses,

since the program searches in the immediate neighborhood ε of the initials. Keeping that in mind, in a

second step we try to depict the loss function from a broader perspective, and understand whether the

minimum loss that was estimated for a reasonable set of parameters in the reaction function represented

a local or a global minimum, and whether the estimated vector of parameters for that global minimum

represented a reasonable set of weights that a policymaker can make use of in his reaction function.

The algorithms are relatively simple to implement, and because the programs are more robust than

many other standard Matlab codes that do approximately the same thing, they have become increas-

ingly popular during the past few years, and are now a standard tool in optimization/welfare loss

analysis in monetary policy theory.

C Annex B

C.1 Figures and Tables

15Zha (2000, 2006)16Which specifies the threshold value at which it proves impossible to improve the function value by more than the

criterion, and hence the iterations will cease. In our simulations, the threshold value was 1 ∗ 10−7

34

Fig

ure

1:E

xte

rnal

finan

cem

arke

t

35

Fig

ure

2:N

etw

orth

offirm

san

dca

pit

alpri

ces

36

Fig

ure

3:F

irm

and

mac

roec

onom

icdat

a

37

Fig

ure

4:H

ouse

hol

dan

dm

acro

econ

omic

dat

a

38

Fig

ure

5:P

ublic

spen

din

gan

dlo

ng-

term

spre

ads

39

Fig

ure

6:R

isk-f

ree

rate

san

dP

rim

era

tesp

read

40

Fig

ure

7:Shor

t-te

rmsp

read

s

41

Tab

le3:

Sec

ond

mom

ents

ofke

ym

acro

econ

omic

vari

able

s

Var

iabl

eP

re-2

000

Post

-2000

Wh

ole

sam

ple

Relativeσ

ρwithoutput

Relativeσ

ρwithoutput

Relativeσ

ρwithoutput

Net

wor

thfi

rms

(his

tori

cal

valu

e)

1.04

0.08

2.18

0.66

1.24

0.21

Net

wor

thfi

rms

(mar

ket

valu

e)1.

41-0

.25

5.25

0.70

2.30

0.07

Net

corp

orat

epr

ofits

5.54

0.58

7.04

0.25

5.76

0.52

S&

P50

0in

dex

pric

ere

turn

s5.

840.

4411

.60.

916.

840.

52

Rea

ldi

spos

able

inco

me

0.74

0.78

0.65

0.52

0.73

0.75

Net

wor

thho

use

hold

s1.

340.

295.

500.

902.

320.

41

Deb

tou

tsta

ndi

ng-

Hou

seho

lds

1.12

0.49

2.28

0.59

1.32

0.49

Deb

tou

tsta

ndi

ng-

Tot

alco

rpor

ate

1.04

0.62

2.56

0.85

1.31

0.63

σσ

σR

eal

GD

P1.

651.

341.

60Str

uct

ura

lB

reak

Tes

tsN

ull

hypo

thes

isH

0:

Datapre−

2000

=Datapost−

2000

HA

:Datapre−

20006=

Datapost−

2000

Box

Bar

tlet

tte

stst

atis

tic

3.69∗

10−

13

p-v

alue:

0

Wal

dte

stst

atis

-ti

c7.

403∗

10−

10

p-v

alue:

0

42

Tab

le4:

Sec

ond

mom

ents

ofke

yfinan

cial

rate

san

dsp

read

s

Var

iabl

eP

re-2

000

Post

-2000

Wh

ole

sam

ple

Relativeσ

ρwithoutput

Relativeσ

ρwithoutput

Relativeσ

ρwithoutput

Fu

nd

rate

1.04

0.17

1.06

0.81

1.04

0.27

T-b

ill

rate

0.72

0.22

0.97

0.78

0.76

0.31

T-b

ond

rate

0.50

-0.0

70.

390.

650.

480.

005

AA

A-s

prea

d0.

15-0

.44

0.29

-0.2

60.

18-0

.39

BA

A-s

prea

d0.

25-0

.63

0.57

-0.4

90.

31-0

.56

Cor

pora

teri

sksp

read

0.14

-0.6

40.

33-0

.62

0.18

-0.5

9

LIB

OR

spre

ad0.

540.

130.

33-0

.15

0.50

0.09

TE

Dsp

read

0.35

-0.0

40.

350.

120.

350

Pri

me

rate

spre

ad0.

36-0

.14

0.19

0.02

0.34

-0.1

3

Pap

er-b

ill

spre

ad0.

50-0

.20

0.28

0.18

0.46

-0.1

8

Str

uct

ura

lB

reak

Tes

tsN

ull

hypo

thes

isH

0:

Datapre−

2000

=Datapost−

2000

HA

:Datapre−

20006=

Datapost−

2000

Box

Bar

tlet

tte

stst

atis

tic

3.69∗

10−

13

p-v

alue:

0

Wal

dte

stst

atis

-ti

c7.

403∗

10−

10

p-v

alue:

0

43

Tab

le5:

Sec

ond

mom

ents

ofot

her

mac

roec

onom

icdat

aV

aria

ble

Pre

-2000

Post

-2000

Wh

ole

sam

ple

Relativeσ

ρwithoutput

Relativeσ

ρwithoutput

Relativeσ

ρwithoutput

Pri

vate

inve

st-

men

t4.

450.

915.

630.

934.

610.

91

Con

sum

ptio

nn

ondu

rabl

es0.

910.

151.

970.

891.

10.

31

TW

EI

(bro

ad)

2.34

-0.1

2.48

-0.5

62.

37-0

.19

Pu

blic

spen

din

g1.

120.

050.

70-0

.64

1.08

-0.0

06In

flat

ion

(GD

P-

defl

ator

)0.

52-0

.58

0.45

0.53

0.50

-0.4

6

σσ

σR

eal

GD

P1.

651.

341.

60Str

uct

ura

lB

reak

Tes

tsN

ull

hypo

thes

isH

0:

Datapre−

2000

=Datapost−

2000

HA

:Datapre−

20006=

Datapost−

2000

Box

Bar

tlet

tte

stst

atis

tic

3.69∗

10−

13

p-v

alue:

0

Wal

dte

stst

atis

-ti

c7.

403∗

10−

10

p-v

alue:

0

44

Table 6: Model parameters

Symbol Parameter ValueC/Y ss steady state consumption share of AD 0.568Ce/Y ss steady state entrepreneurial consumption share of AD 0.04I/Y ss steady state investment share of AD 0.192G/Y ss steady state government spending share of AD 0.2Rkss steady state return on capital 1.025K/N steady state leverage ratio of entrepreneurs 2β quarterly discount factor 0.99σ intertemporal consumption elasticity 0.1φ elasticity of price of capital to I/K ratio 1δ quarterly depreciation rate of capital 0.025ψ elasticity of external finance premium to leverage 0.05α capital share in production 0.35χ parameter on net wealth accumulation 1.021γ survival rate of entrepreneurs 0.9728γf coefficient on forward looking inflation in Phillips curve 0.5γb coefficient on backward looking inflation in Phillips curve 0.5γl inverse of labor supply elasticity (= 1 + 1/εls) 1.33γc coefficient on consumption in labor market equation 10γπ coefficient on inflation in Taylor rule 1.01 and 2κ parameter on marginal cost in Phillips curve 0.086ε parameter on marginal product in investment demand 0.95b parameter on the evolution of the bubble 0.9533

45

Figure 8: Productivity shock

46

Figure 9: Aggregate demand shock

47

Figure 10: Monetary policy shock

48

Figure 11: Asset price bubble shock

49

Table 7: Model second momentsVariable Relative stnd deviation Correlation to output

γπ = 1.4 γπ = 2.4 γπ = 1.4 γπ = 2.4

Investment 4.27 4.30 1 1

Consump. 0.06 0.06 0.05 -0.15

Net worth 4.60 4.50 0.95 0.93

Leverage 3.14 2.97 0.91 0.89

Market asset price 2.44 2.41 1 0.99

Price of fundamental 0.66 0.59 0.51 0.39

Capital 0.15 0.15 0.32 0.32

Marginal cost 1.31 1.19 0.85 0.83

Finance premium 0.58 0.56 -0.30 -0.24

Nominal rate 0.54 0.57 0.08 0.16

Inflation 0.21 0.17 0.63 0.59

Savings rate 0.58 0.56 -0.1 0.02Standard deviation

Output 0.50 0.48

50

Tab

le8:

Wel

fare

anal

ysi

sof

alte

rnat

ive

mon

etar

yp

olic

ies

σ=

1A

llfo

ur

shock

s’s

wit

ched

on’

Rea

ctio

nfu

nct

ion

Init

ial

wei

ghts

-rea

ctio

nfu

nct

ion

Val

ue

of’b

’L

oss-

valu

eO

pti

mal

wei

ghts

(Et(πt+

1))

(2)

0.98∗

0.97

50.

0051

(0.9

4)(E

t(πt+

1),y t

)(2,0.5

)0.

98∗

0.97

50.

0033

(−1.

17,1.3

3)(E

t(πt+

1),y t,qt)

(2,0.5,0.2

)0.

98∗

0.97

50.

0052

(−1.

8,0.

71,0.4

1)(E

t(πt+

1),y t,st)

(2,0.5,0.2

)0.

98∗

0.97

50.

0026

(−1.

78,0.7

2,0.

42)

(Et(πt+

1),y t,bub t

)(2,0.5,0.2

)0.

98∗

0.97

50.

0026

(−1.

87,0.6

3,0.

33)

(Et(πt+

1))

(2)

1∗

0.97

50.

0049

(0.9

4)(E

t(πt+

1),y t

)(2,0.5

)1∗

0.97

50.

0032

(−1.

168,

1.33

)(E

t(πt+

1),y t,qt)

(2,0.5,0.2

)1∗

0.97

50.

0052

(−1.

8,0.

70,0.4

0)(E

t(πt+

1),y t,st)

(2,0.5,0.2

)1∗

0.97

50.

0026

(−1.

77,0.7

3,0.

43)

(Et(πt+

1),y t,bub t

)(2,0.5,0.2

)1∗

0.97

50.

0025

(−1.

88,0.6

8,0.

33)

(Et(πt+

1))

(2)

1.02

5641

03∗

0.97

50.

0076

(0.7

6)(E

t(πt+

1),y t

)(2,0.5

)1.

0255

4103∗

0.97

50.

0067

(1.2

6,1.

24)

(Et(πt+

1),y t,qt)

(2,0.5,0.2

)1.

0256

4103∗

0.97

50.

0080

(1.8

8,0.

62,0.3

2)(E

t(πt+

1),y t,st)

(2,0.5,0.2

)1.

0256

4103∗

0.97

50.

0068

(1.8

3,0.

67,0.3

7)(E

t(πt+

1),y t,bub t

)(2,0.5,0.2

)1.

0256

4103∗

0.97

50.

0067

(1.9

2,0.

58,0.2

8)C

ontr

olte

sts

Rea

ctio

nfu

nct

ion

Init

ial

wei

ghts

-rea

ctio

nfu

nct

ion

Val

ue

of’b

’L

oss-

valu

eO

pti

mal

wei

ghts

(Et(πt+

1),y t

)(3,0.5

)0.

98∗

0.97

50.

0030

(−1.

63,1.8

7)(E

t(πt+

1),y t,qt)

(3,0.5,0.2

)0.

98∗

0.97

50.

0049

(−2.

57,0.9

3,0.

63)

(Et(πt+

1),y t,st)

(3,0.5,0.2

)0.

98∗

0.97

50.

0026

(−2.

70,0.8

2,0.

52)

(Et(πt+

1),y t,bub t

)(3,0.5,0.2

)0.

98∗

0.97

50.

0025

(−2.

81,0.6

9,0.

39)

(Et(πt+

1),Et(y t

+1))

(2,0.5

)0.

98∗

0.97

50.

0048

(−0.

76,0.7

4)(E

t(πt+

1),Et(y t

+1),q t

)(2,0.5,0.2

)0.

98∗

0.97

50.

0052

(−2.

37,−

0.87,−

0.17

)(E

t(πt+

1),Et(y t

+1),s t

)(2,0.5,0.2

)0.

98∗

0.97

50.

0027

(−1.

76,−

0.26,0.4

4)(E

t(πt+

1),Et(y t

+1),bub t

)(2,0.5,0.2

)0.

98∗

0.97

50.

0026

(−1.

85,−

0.35,0.3

5)

51

Table 9: Loss function analysis 1b=0.98*(1-0.025)

Initial weights Reaction function Loss function Optimal weights(2) (Et(πt+1)) 0.0051 (0.94)(20) (Et(πt+1)) 0.0051 (44.03)(40) (Et(πt+1)) 0.0051 (44.03)(2,0.5) (Et(πt+1), yt) 0.0032 (1.17,1.33)(2,0.01) (Et(πt+1), yt) 0.0037 (0.97,1.04)(2,0.1) (Et(πt+1), yt) 0.0036 (1,1.10)(2,1) (Et(πt+1), yt) 0.0031 (1.39,1.61)(2,2) (Et(πt+1), yt) 0.0029 (1.88,2.12)(1,0.5) (Et(πt+1), yt) 0.0044 (0.78,0.72)(1.5,0.5) (Et(πt+1), yt) 0.0037 (0.97,1.04)(3,0.5) (Et(πt+1), yt) 0.0030 (1.63,1.87)(4,0.5) (Et(πt+1), yt) 0.0028 (2.14,2.36)(10,0.5) (Et(πt+1), yt) 0.0027 (5.89,4.61)(20,0.5) (Et(πt+1), yt) 0.0028 (13.32,7.19)(0.5,0.5) (Et(πt+1), yt) 0.0061 (0.62,0.38)(5,5) (Et(πt+1), yt) 0.0027 (5.55,4.45)(2,0.5,0.2) (Et(πt+1), yt, qt) 0.0052 (1.79,0.71,0.41)(2,0.5,0.01) (Et(πt+1), yt, qt) 0.0047 (1.68,0.82,0.33)(2,0.5,0.1) (Et(πt+1), yt, qt) 0.0049 (1.73,0.77,0.37)(2,0.5,0.3) (Et(πt+1), yt, qt) 0.0056 (1.85,0.66,0.46)(2,0.5,1) (Et(πt+1), yt, qt) 0.0079 (2.18,0.33,0.83)(2,0.1,0.2) (Et(πt+1), yt, qt) 0.0070 (1.70,0.41,0.51)(2,0.4,0.2) (Et(πt+1), yt, qt) 0.0056 (1.77,0.64,0.44)(2,0.45,0.2) (Et(πt+1), yt, qt) 0.0054 (1.78,0.67,0.42)(2,0.55,0.2) (Et(πt+1), yt, qt) 0.0051 (1.80,0.75,0.40)(2,0.6,0.2) (Et(πt+1), yt, qt) 0.0049 (1.81,0.79,0.39)(2,1,0.2) (Et(πt+1), yt, qt) 0.0039 (1.89,1.11,0.31)

52

Table 10: Loss function analysis 2b=0.98*(1-0.025) cont.

Initial weights Reaction function Loss function Optimal weights(1,0.5,0.2) (Et(πt+1), yt, qt) 0.0058 (1.03,0.47,0.17)(1.5,0.5,0.2) (Et(πt+1), yt, qt) 0.0055 (1.41,0.59,0.29)(2.5,0.5,0.2) (Et(πt+1), yt, qt) 0.0051 (2.18,0.82,0.52)(3,0.5,0.2) (Et(πt+1), yt, qt) 0.0049 (2.57,0.93,0.63)(2,0.5,0.2) (Et(πt+1), yt,∆t) 0.0026 (1.78,0.72,0.42)(2,0.5,0.01) (Et(πt+1), yt,∆t) 0.0026 (1.63,0.87,0.38)(2,0.5,0.1) (Et(πt+1), yt,∆t) 0.0026 (1.70,0.80,0.40)(2,0.5,0.25) (Et(πt+1), yt,∆t) 0.0026 (1.81,0.69,0.43)(2,0.5,0.3) (Et(πt+1), yt,∆t) 0.0027 (1.85,0.65,0.45)(2,0.5,0.5) (Et(πt+1), yt,∆t) 0.0027 (2.01,0.49,0.49)(2,0.1,0.2) (Et(πt+1), yt,∆t) 0.0028 (1.73,0.37,0.47)(2,1,0.2) (Et(πt+1), yt,∆t) 0.0025 (1.83,1.17,0.37)(2,2,0.2) (Et(πt+1), yt,∆t) 0.0025 (1.91,2.09,0.29)(1,0.5,0.2) (Et(πt+1), yt,∆t) 0.0031 (0.95,0.55,0.25)(1.5,0.5,0.2) (Et(πt+1), yt,∆t) 0.0028 (1.35,0.65,0.35)(2.5,0.5,0.2) (Et(πt+1), yt,∆t) 0.0026 (2.23,0.78,0.48)(3,0.5,0.2) (Et(πt+1), yt,∆t) 0.0026 (2.69,0.82,0.52)(8,0.5,0.2) (Et(πt+1), yt,∆t) 0.0027 (7.46,1.04,0.74)(2,0.5,0.2) (Et(πt+1), yt, bubt) 0.0026 (1.81,0.63,0.33)(2,0.5,0.00001) (Et(πt+1), yt, bubt) 0.0026 (1.71,0.80,0.30)(2,0.5,0.01) (Et(πt+1), yt, bubt) 0.0026 (1.71,0.79,0.30)(2,0.5,0.1) (Et(πt+1), yt, bubt) 0.0026 (1.71,0.71,0.31)(2,0.5,0.3) (Et(πt+1), yt, bubt) 0.0026 (1.96,0.55,0.35)(2,0.5,0.5) (Et(πt+1), yt, bubt) 0.0026 (2.12,0.38,0.38)(2,0.5,8) (Et(πt+1), yt, bubt) 0.0027 (12.63,2.56,0.47)(2,0.5,10) (Et(πt+1), yt, bubt) 0.0028 (17.78,5.07,0.29)(2,0.01,0.2) (Et(πt+1), yt, bubt) 0.0026 (1.83,0.18,0.37)(2,0.1,0.2) (Et(πt+1), yt, bubt) 0.0026 (1.84,0.27,0.37)(2,1,0.2) (Et(πt+1), yt, bubt) 0.0026 (1.92,1.08,0.28)(2,2,0.2) (Et(πt+1), yt, bubt) 0.0027 (2.03,1.97,0.17)(2,10,0.2) (Et(πt+1), yt, bubt) 0.0033 (3.17,8.83,-0.97)(0.5,0.5,0.2) (Et(πt+1), yt, bubt) 0.0047 (0.61,0.39,0.09)(1,0.5,0.2) (Et(πt+1), yt, bubt) 0.0030 (0.98,0.52,0.22)(1.5,0.5,0.2) (Et(πt+1), yt, bubt) 0.0027 (1.41,0.59,0.29)(2.2,0.5,0.2) (Et(πt+1), yt, bubt) 0.0025 (2.06,0.64,0.34)(3,0.5,0.2) (Et(πt+1), yt, bubt) 0.0025 (2.81,0.69,0.39)(5.4,0.5,0.2) (Et(πt+1), yt, bubt) 0.0026 (5.12,0.78,0.48)

53

Table 11: Loss function analysis 3b=1*(1-0.025)

Initial weights Reaction function Loss function Optimal weights(2) Et(πt+1) 0.0049 (0.94)(10) Et(πt+1) 0.0049 (44.03)(40) Et(πt+1) 0.0049 (44.03)(2,0.5) Et(πt+1, yt) 0.0032 (1.17,1.33)(2,0.1) Et(πt+1, yt) 0.0035 (1,1.10)(2,0.3) Et(πt+1, yt) 0.0034 (1.09,1.22)(2,0.75) Et(πt+1, yt) 0.0031 (1.28,1.97)(2,1) Et(πt+1, yt) 0.0030 (1.39,1.61)(1,0.5) Et(πt+1, yt) 0.0043 (0.78,0.72)(1.5,0.5) Et(πt+1, yt) 0.0036 (0.97,1.04)(1.9,0.5) Et(πt+1, yt) 0.0033 (1.13,1.27)(2.5,0.5) Et(πt+1, yt) 0.0030 (1.39,1.61)(3,0.5) Et(πt+1, yt) 0.0029 (1.63,1.87)(4,0.5) Et(πt+1, yt) 0.0027 (2.14,2.36)(5,0.5) Et(πt+1, yt) 0.0026 (2.69,2.81)(10,0.5) Et(πt+1, yt) 0.0026 (5.89,4.61)(2,0.5,0.2) Et(πt+1, yt, qt) 0.0052 (1.81,0.70,0.40)(2,0.5,0.01) Et(πt+1, yt, qt) 0.0046 (1.70,0.80,0.81)(2,0.5,0.1) Et(πt+1, yt, qt) 0.0049 (1.75,0.75,0.35)(2,0.5,0.25) Et(πt+1, yt, qt) 0.0054 (1.83,0.67,0.42)(2,0.5,0.3) Et(πt+1, yt, qt) 0.0055 (1.86,0.64,0.44)(2,0.5,0.5) Et(πt+1, yt, qt) 0.0062 (1.96,0.55,0.55)(2,0.1,0.2) Et(πt+1, yt, qt) 0.0070 (1.70,0.40,0.30)(2,0.3,0.2) Et(πt+1, yt, qt) 0.0060 (1.76,0.54,0.44)(2,0.4,0.2) Et(πt+1, yt, qt) 0.0056 (1.78,0.62,0.42)(2,0.6,0.2) Et(πt+1, yt, qt) 0.0049 (1.83,0.77,0.37)(2,0.75,0.2) Et(πt+1, yt, qt) 0.0044 (1.86,0.89,0.234(2,1,0.2) Et(πt+1, yt, qt) 0.0039 (1.92,1.09,0.29)

54

Table 12: Loss function analysis 4b=1*(1-0.025) cont.

Initial weights Reaction function Loss function Optimal weights(1,0.5,0.2) Et(πt+1, yt, qt) 0.0057 (1.04,0.46,0.16)(1.5,0.5,0.2) Et(πt+1, yt, qt) 0.0054 (1.42,0.58,0.28)(1.75,0.5,0.2) Et(πt+1, yt, qt) 0.0053 (1.61,0.64,0.34)(2.5,0.5,0.2) Et(πt+1, yt, qt) 0.0050 (2.20,0.81,0.51)(3,0.5,0.2) Et(πt+1, yt, qt) 0.0049 (2.59,0.91,0.61)(4,0.5,0.2) Et(πt+1, yt, qt) 0.0047 (3.38,1.12,0.81)(10,0.5,0.2) Et(πt+1, yt, qt) 0.0042 (8.32,2.18,1.88)(20,0.5,0.2) Et(πt+1, yt, qt) 0.0040 (16.89,3.61,3.31)(2,0.5,0.2) Et(πt+1, yt,∆t) 0.0026 (1.77,0.73,0.43)(2,0.5,0.01) Et(πt+1, yt,∆t) 0.0025 (1.63,0.88,0.39)(2,0.5,0.1) Et(πt+1, yt,∆t) 0.0026 (1.70,0.81,0.41)(2,0.5,0.3) Et(πt+1, yt,∆t) 0.0026 (1.85,0.65,0.45)(2,0.5,0.4) Et(πt+1, yt,∆t) 0.0026 (1.93,0.57,0.47)(2,0.5,0.7) Et(πt+1, yt,∆t) 0.0027 (2.16,0.34,0.54)(2,0.5,1) Et(πt+1, yt,∆t) 0.0027 (2.40,0.11,0.61)(2,0.1,0.2) Et(πt+1, yt,∆t) 0.0027 (1.73,0.37,0.47)(2,0.6,0.2) Et(πt+1, yt,∆t) 0.0025 (1.78,0.82,0.42)(2,1,0.2) Et(πt+1, yt,∆t) 0.0025 (1.82,1.18,0.38)(2,2,0.2) Et(πt+1, yt,∆t) 0.0024 (1.91,2.09,0.29)(2,3,0.2) Et(πt+1, yt,∆t) 0.0026 (1.99,3.01,0.21)(1,0.5,0.2) Et(πt+1, yt,∆t) 0.0031 (0.95,0.55,0.25)(1.5,0.5,0.2) Et(πt+1, yt,∆t) 0.0027 (1.34,0.66,0.36)(2.5,0.5,0.2) Et(πt+1, yt,∆t) 0.0025 (2.22,0.78,0.48)(3,0.5,0.2) Et(πt+1, yt,∆t) 0.0025 (2.68,0.82,0.52)(7,0.5,0.2) Et(πt+1, yt,∆t) 0.0026 (6.48,1.02,0.72)(2,0.5,0.2) Et(πt+1, yt, bubt) 0.0025 (1.87,0.63,0.33)(2,0.5,0.01) Et(πt+1, yt, bubt) 0.0025 (1.72,0.78,0.29)(2,0.5,0.1) Et(πt+1, yt, bubt) 0.0025 (1.79,0.71,0.31)(2,0.5,0.5) Et(πt+1, yt, bubt) 0.0025 (2.13,0.37,0.37)(2,0.5,1) Et(πt+1, yt, bubt) 0.0026 (2.56,0.06,0.45)(2,0.1,0.2) Et(πt+1, yt, bubt) 0.0025 (1.84,0.26,0.36)(2,1,0.2) Et(πt+1, yt, bubt) 0.0025 (1.92,1.08,0.28)(2,1.7,0.2) Et(πt+1, yt, bubt) 0.0026 (2,1.70,0.20)(1,0.5,0.2) Et(πt+1, yt, bubt) 0.0029 (0.98,0.52,0.22)(1.5,0.5,0.2) Et(πt+1, yt, bubt) 0.0026 (1.42,0.58,0.28)(2.8,0.5,0.2) Et(πt+1, yt, bubt) 0.0024 (2.63,0.68,0.38)(4,0.5,0.2) Et(πt+1, yt, bubt) 0.0025 (3.77,0.73,0.43)

55

Table 13: Loss function analysis 5b=1.02564103*(1-0.025)

Initial weights Reaction function Loss function Optimal weights(2) Et(πt+1) 0.0076 (0.76)(4) Et(πt+1) 0.0076 (0.76)(20) Et(πt+1) 0.0078 (56.58)(2,0.5) Et(πt+1, yt) 0.0067 (1.26,1.24)(2,0.1) Et(πt+1, yt) 0.0068 (1.02,1.08)(2,1) Et(πt+1, yt) 0.0067 (1.6,1.4)(2,2) Et(πt+1, yt) 0.0068 (2.39,1.61)(2,4) Et(πt+1, yt) 0.0069 (4.2,1.8)(2,5) Et(πt+1, yt) 0.0070 (5.15,1.86)(3,0.1) Et(πt+1, yt) 0.0067 (1.67,1.43)(1,0.5) Et(πt+1, yt) 0.0071 (0.74,0.76)(0.5,0.5) Et(πt+1, yt) 0.0080 (0.59,0.41)(3,0.5) Et(πt+1, yt) 0.0067 (1.98,1.52)(4,0.5) Et(πt+1, yt) 0.0068 (2.82,1.68)(5,0.5 Et(πt+1, yt) 0.0069 (3.73,1.77)(3,3) Et(πt+1, yt) 0.0069 (4.2,1.8)(5,5) Et(πt+1, yt) 0.0071 (8.05,1.95)(2,0.5,0.2) Et(πt+1, yt, qt) 0.0080 (1.88,0.62,0.32)(2,0.5,0.001) Et(πt+1, yt, qt) 0.0076 (1.78,0.72,0.22)(2,0.5,0.01) Et(πt+1, yt, qt) 0.0077 (1.78,0.72,0.23)(2,0.5,0.1) Et(πt+1, yt, qt) 0.0078 (1.83,0.67,0.27)(2,0.01,0.2) Et(πt+1, yt, qt) 0.0094 (1.75,0.26,0.45)(2,0.1,0.2) Et(πt+1, yt, qt) 0.0091 (1.77,0.32,0.42)(2,1,0.2) Et(πt+1, yt, qt) 0.0072 (2.07,0.93,0.13)(2,2,0.2) Et(πt+1, yt, qt) 0.0068 (2.08,1.92,0.12)(2,3,0.2) Et(πt+1, yt, qt) 0.0072 (1.76,3.24,0.44)(2,5,0.2) Et(πt+1, yt, qt) 0.0085 (1.47,5.53,0.73)(0.5,0.5,0.2) Et(πt+1, yt, qt) 0.0083 (0.67,0.33,0.03)(1,0.5,0.2) Et(πt+1, yt, qt) 0.0082 (1.08,0.42,0.12)(3,0.5,0.2) Et(πt+1, yt, qt) 0.0079 (2.70,0.81,0.51)(5,0.5,0.2) Et(πt+1, yt, qt) 0.0078 (4.38,1.12,0.82)(7,0.5,0.2) Et(πt+1, yt, qt) 0.0077 (6.12,1.38,1.08)(2,2,0.1) Et(πt+1, yt, qt) 0.0069 (1.79,2.12,0.21)

56

Table 14: Loss function analysis 6b=1.0256*(1-0.025) cont.

Initial weights Reaction function Loss function Optimal weights(2,0.5,0.2) Et(πt+1, yt,∆t) 0.0068 (1.83,0.67,0.37)(2,0.5,0.001) Et(πt+1, yt,∆t) 0.0068 (1.70,0.80,0.30)(2,0.5,1) Et(πt+1, yt,∆t) 0.0071 (2.34,0.16,0.66)(2,0.5,2) Et(πt+1, yt,∆t) 0.0073 (2,99.49,1.01)(2,2,0.2) Et(πt+1, yt,∆t) 0.0068 (2.28,1.73,0.08)(2,4,0.2) Et(πt+1, yt,∆t) 0.0070 (2.94,3.06,0.74)(0.5,0.5,0.2) Et(πt+1, yt,∆t) 0.0077 (0.63,0.37,0.07)(1,0.5,0.2) Et(πt+1, yt,∆t) 0.0070 (0.98,0.52,0.22)(3,0.5,0.2) Et(πt+1, yt,∆t) 0.0069 (2.76,0.74,0.44)(5,0.5,0.2) Et(πt+1, yt,∆t) 0.0070 (4.71,0.79,0.49)(2,2,0.001) Et(πt+1, yt,∆t) 0.0068 (2.15,1.85,0.15)(2,0.5,0.2) Et(πt+1, yt, bubt) 0.0067 (1.92,0.58,0.28)(2,0.5,0.001) Et(πt+1, yt, bubt) 0.0067 (1.77,0.73,0.23)(2,0.5,1) Et(πt+1, yt, bubt) 0.0067 (2.51,0.005,0.5)(2,0.5,2) Et(πt+1, yt, bubt) 0.0068 (3.25,0.75,0.75)(2,0.5,4) Et(πt+1, yt, bubt) 0.0069 (4.74,2.24,1.26)(2,0.01,0.2) Et(πt+1, yt, bubt) 0.0067 (1.81,0.2,0.39)(2,2,0.2) Et(πt+1, yt, bubt) 0.0068 (2.25,1.75,0.05)(2,4,0.2) Et(πt+1, yt, bubt) 0.0069 (2.71,3.29,0.51)(0.5,0.5,0.2) Et(πt+1, yt, bubt) 0.0073 (0.6,0.4,0.1)(1,0.5,0.2) Et(πt+1, yt, bubt) 0.0067 (1,0.5,0.2)(3,0.5,0.2) Et(πt+1, yt, bubt) 0.0068 (2.88,0.62,0.32)(5,0.5,0.2) Et(πt+1, yt, bubt) 0.0069 (4.84,0.66,0.36)(2,0.01,0.01) Et(πt+1, yt, bubt) 0.0067 (1.67,0.34,0.34)

57

Table 15: Loss function analysis 7b=1.0256*(1-0.025) cont.

Initial weights Reaction function Loss function Optimal weights(2) Et(πt+1) 0.0048 (0.95)(4) Et(πt+1) 0.0048 (0.95)(10) Et(πt+1) 0.0048 (44.03)(2,0.5) Et(πt+1, yt) 0.0031 (1.17,1.33)(2,0.1) Et(πt+1, yt) 0.0034 (1,1.1)(2,0.3) Et(πt+1, yt) 0.0033 (1.1,1.22)(2,0.4) Et(πt+1, yt) 0.0032 (1.13,1.27)(2,1) Et(πt+1, yt) 0.0029 (1.39,1.61)(2,2) Et(πt+1, yt) 0.0026 (1.88,2.13)(2,4) Et(πt+1, yt) 0.0025 (2.99,3.02)(1,0.5) Et(πt+1, yt) 0.0042 (0.78,0.72)(1.5,0.5) Et(πt+1, yt) 0.0035 (0.97,1.04)(1.75,0.5) Et(πt+1, yt) 0.0033 (1.06,1.19)(2.5,0.5) Et(πt+1, yt) 0.0029 (1.39,1.61)(3,0.5 Et(πt+1, yt) 0.0028 (1.63,1.87)(4,0.5) Et(πt+1, yt) 0.0026 (2.14,2.36)(6,0.5) Et(πt+1, yt) 0.0025 (3.28,3.22)(2,0.5,0.2) Et(πt+1, yt, qt) 0.0052 (1.83,0.68,0.38)(2,0.5,0.01) Et(πt+1, yt, qt) 0.0046 (1.72,0.78,0.29)(2,0.5,0.1) Et(πt+1, yt, qt) 0.0048 (1.77,0.73,0.33)(2,0.5,0.5) Et(πt+1, yt, qt) 0.0062 (1.97,0.53,0.53)(2,0.1,0.2) Et(πt+1, yt, qt) 0.0070 (1.72,0.39,0.49)(2,0.4,0.2) Et(πt+1, yt, qt) 0.0056 (1.80,0.60,0.40)(2,1,0.2) Et(πt+1, yt, qt) 0.0038 (1.95,1.05,0.25)(2,2,0.2) Et(πt+1, yt, qt) 0.0027 (2.01,1.99,0.19)(2,3,0.2) Et(πt+1, yt, qt) 0.0025 (1.80,3.2,0.40)(2,7,0.2) Et(πt+1, yt, qt) 0.0026 (1.13,7.87,1.07)(1,0.5,0.2) Et(πt+1, yt, qt) 0.0056 (1.05,0.45,0.15)(3,0.5,0.2) Et(πt+1, yt, qt) 0.0049 (2.62,0.88,0.58)(5,0.5,0.2) Et(πt+1, yt, qt) 0.0046 (4.23,1.28,0.98)(10,0.5,0.2) Et(πt+1, yt, qt) 0.0025 (5.46,9.08,1.28)(2,0.5,0.2) Et(πt+1, yt,∆t) 0.0025 (1.77,0.73,0.43)(2,0.5,0.01) Et(πt+1, yt,∆t) 0.0025 (1.62,0.88,0.39)(2,0.5,0.1) Et(πt+1, yt,∆t) 0.0025 (1.69,0.81,0.41)(2,0.5,0.5) Et(πt+1, yt,∆t) 0.0025 (2,0.50,0.50)(2,0.1,0.2) Et(πt+1, yt,∆t) 0.0027 (1.72,0.38,0.48)(2,0.6,0.2) Et(πt+1, yt,∆t) 0.0025 (1.77,0.82,0.42)(2,0.7,0.2) Et(πt+1, yt,∆t) 0.0024 (1.79,0.91,0.41)(2,1,0.2) Et(πt+1, yt,∆t) 0.0024 (1.82,1.18,0.38)(2,3,0.2) Et(πt+1, yt,∆t) 0.0025 (1.98,3.02,0.22)(1,0.5,0.2) Et(πt+1, yt,∆t) 0.0030 (0.95,0.55,0.25)(1.5,0.5,0.2) Et(πt+1, yt,∆t) 0.0026 (1.34,0.66,0.36)(3,0.5,0.2) Et(πt+1, yt,∆t) 0.0024 (2.67,0.83,0.53)(6,0.5,0.2) Et(πt+1, yt,∆t) 0.0025 (5.51,0.99,0.69)(2,0.5,0.2) Et(πt+1, yt, bubt) 0.0024 (1.88,0.62,0.32)(2,0.5,0.15) Et(πt+1, yt, bubt) 0.0024 (1.84,0.66,0.31)(2,0.5,0.5) Et(πt+1, yt, bubt) 0.0024 (2.13,0.37,0.37)(2,0.5,0.9) Et(πt+1, yt, bubt) 0.0025 (2.47,0.03,0.43)

58